grade 8 | unit 4 systems of linear equations
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STUDY GUIDE
GRADE 8 | UNIT 4
Systems of Linear Equations
Table of Contents
Introduction ............................................................................................................................................... 3
Test Your Prerequisite Skills ........................................................................................................ 4
Objectives ...................................................................................................................................... 5
Lesson 1: Introduction to Systems of Linear Equations
- Warm Up! ........................................................................................................................... 5
- Learn about It! ................................................................................................................... 6
- Let’s Practice! ..................................................................................................................... 8
- Check Your Understanding! ............................................................................................ 12
Lesson 2: Solution of a System of Linear Equations in Two Variables
- Warm Up! ......................................................................................................................... 13
- Learn about It! ................................................................................................................. 14
- Let’s Practice! ................................................................................................................... 17
- Check Your Understanding! ............................................................................................ 22
Lesson 3: Solving Systems of Linear Equations in Two Variables: Substitution
- Warm Up! ......................................................................................................................... 23
- Learn about It! ................................................................................................................. 24
- Let’s Practice! ................................................................................................................... 25
- Check Your Understanding! ............................................................................................ 31
Lesson 4: Solving Systems of Linear Equations in Two Variables: Elimination
- Warm Up! ......................................................................................................................... 32
- Learn about It! ................................................................................................................. 33
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- Let’s Practice! ................................................................................................................... 34
- Check Your Understanding! ............................................................................................ 41
Lesson 5: Solving Systems of Linear Equations in Two Variables: Graphing
- Warm Up! ......................................................................................................................... 42
- Learn about It! ................................................................................................................. 43
- Let’s Practice! ................................................................................................................... 46
- Check Your Understanding! ............................................................................................ 55
Challenge Yourself! ..................................................................................................................... 55
Performance Task ....................................................................................................................... 56
Wrap-up ....................................................................................................................................... 59
Key to Let’s Practice! .................................................................................................................... 61
References ................................................................................................................................... 62
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STUDY GUIDE
GRADE 8 | MATHEMATICS
UNIT 4
Systems of Linear Equations
Have you ever encountered solving an equation where two variables are not only bound
by a single constraint but by two or more? Chances are you are trying to solve a system of
linear equations.
Systems of linear equations can be illustrated in
varied real-life situations. Its usefulness and
versatility is used extensively in various sciences and
disciplines.
Often times, we encounter real-life problems where one quantity is dependent on
another quantity, but that these quantities are bound by not just a single constraint.
Uniform motion problems, mixture and work
problems, investment problems, and even
puzzle problems are examples of the
application of systems of linear equations.
This unit shall discuss how our day-to-day constraints may be
modeled mathematically through a system of linear equations. The
underlying principles regarding the solutions to these problems are
systematically discussed in this unit. Algebraic and graphical
methods to illustrate and solve these systems are discussed in
different lessons in this unit.
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Table of Contents
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Before you get started, answer the following items to help you assess your prior
knowledge and practice some skills that you will need in studying the lessons in this unit.
1. Determine which of the following ordered pairs are solutions to the equation
.
a. (0, 15) b. (1, 11) c. (4, 3) d. (5, 4)
2. Determine the value of in for a given value of .
a. b. c.
d.
3. Transform the following equations to slope-intercept form.
a. b. c. d.
4. Complete the table of values for each given equation then graph.
a. b.
Determining whether or not an ordered pair is a solution to a linear
equation
Solving for the value of a variable in a linear equation given the value of
the other variable
Transforming linear equations from standard form to slope-intercept
form
Graphing linear equations
Modeling word problems as linear equations
Test Your Prerequisite Skills
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5. Write a linear equation that would model the problem below and solve it:
Pampanga and Manila are about 240 km apart. A car leaves Manila traveling
towards Pampanga at 65 kph. At the same time, a bus leaves Pampanga bound for
Manila at 55 kph. How long will it take before they meet?
At the end of this unit, you should be able to recognize systems of linear equations;
identify if a point is a solution to a system of linear equations or not;
enumerate the different types of solutions of a system of linear equations in two
variables;
define each category which a system of linear equation could fall into;
identify, without graphing, which system of linear equations has parallel,
intersecting, or overlapping lines; and
solve a system of linear equations in two variables by substitution, graphing, and
elimination.
Picture It Clearly!
Materials Needed: activity sheet, ruler, coloring and drawing materials
Instructions:
1. The activity may be done by pair or by group.
Lesson 1: Introduction to Systems of Linear Equations
Objectives
Warm Up!
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2. Your teacher will provide an activity sheet in the form of an image half the
size of a legal bond paper.
3. The image or figure may be in the form of a work of art, a landscape,
infrastructure, etc.
4. Construct a rectangular coordinate plane on the image and label it
accordingly.
5. From the image, find elements that illustrate each mathematical concept
below.
lines points
on a line
parallel
lines
slope of a
line points
intersecting
lines abscissa
graph of
a line
coordinates
of points
linear
equations
6. Describe each concept using the format given below.
“My idea of (state the mathematical concept) is
________________________________”
Some of the terms you encountered and illustrated in the Warm Up! activity, words like
graph of a line, linear equations, slope of a line, and points, will recurrently be used in this
lesson and the succeeding ones. Reviewing prior knowledge on these will be helpful in
understanding the concepts in this unit.
Let us now discuss what a system of linear equations is.
Learn about It!
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A pair of lines whether parallel, intersecting, or coinciding illustrates a system of linear
equations.
The degree of the variables in linear equations is either 0 or 1 and no two variables are
being multiplied with each other. So, if you see or in the equation, then it is not
linear.
The graph of a system of linear equations will involve two or more lines.
Consider the following system of linear equations in two variables and its graph:
Definition 1.1: A system of linear equations involves two or
more linear equations.
It may have the form
, , , , and are constants, while and ,
and and , must not be both equal to zero.
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A solution to a system of linear equations in two variables is an ordered pair for
which the values of and satisfy each of the equations in the system.
Graphically, these two values are represented as the point of intersection, with ordered
pair , of both lines.
Example 1: Give an example of a system of linear equations in one variable.
Solution: Answers may vary but since the instruction says the equation needs to be in
one variable, we can use either or . Let us say, for example, we use .
One possible answer would be
Note that the variable in both equations is of degree 1 or 0 (specifically,
both are of degree 1).
Try It Yourself!
Give an example of a system of linear equations in two variables.
Definition 1.2: The point of intersection of two lines is
the point where the two lines meet.
Let’s Practice!
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Example 2: Show that the point ( is a solution of the given system of linear equations
in two variables.
Solution:
Step 1: Identify the values of and from the given point.
Given the point , and .
Step 2: Substitute the values of and into each of the equations in the system, and
then simplify.
Equation 1:
The point satisfies the first equation.
Equation 2:
The point satisfies the second equation.
Therefore, the point is the solution of the system
since it
satisfies both equations.
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Try It Yourself!
Is the point a solution of the given system of linear equations in two
variables?
Example 3: Determine whether is a solution of the system of equations
and .
Solution:
Step 1: Identify the values of and from the given point.
Given the point , and .
Step 2: Substitute the values of and into each of the equations in the system, and
then simplify.
Equation 1:
The point satisfies the first equation.
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Equation 2:
The point satisfies the second equation.
Therefore, the point (3, 2) is the solution of the system
since it
satisfies both equations.
Try It Yourself!
Determine whether or not is a solution of the system of equations
and .
Real-World Problems
Example 4: An exam contains 20 questions. Some items are worth 2
points and the rest are worth 3 points. The exam is worth
100 points. Construct a system of linear equations that
models the situation.
Solution:
Step 1: Define the variables.
Let be the number of two-point questions.
Let be the number of three-point questions.
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Step 2: Translate English phrases or statements into mathematical expressions or
equations.
An exam contains 20 questions. ⇒
Some items are worth 2 points ⇒
The rest are worth 3 points ⇒
The exam is worth 100 points. ⇒
Therefore, the system of linear equations that models the situation is
.
Try It Yourself!
A jeepney driver charges ₱8 for passengers who rode for
less than 4 km and ₱10 for passengers who rode for
more than 4 km. The jeepney has a total capacity of 20
passengers, which makes him earn ₱180. Construct a
system of linear equations that models the situation
where is the number of passengers who rode for less than 4 km and is the
number of passengers who rode for more than 4 km.
1. Give two examples of a system of linear equations in two variables.
2. Determine if the point is a solution to the system of linear equations
Check Your Understanding!
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3. The admission fee at a theme park is ₱300 for children and ₱500 for adults. A
family of ten entered the theme park and paid ₱2,600. Construct a system of linear
equations that represent the problem.
What Do They Look Like?
Materials Needed: graphing paper, ruler, pencil or any drawing materials
Instructions:
1. The activity may be done by groups of three or five.
2. Your group will be given a system of linear equations, and you are to graph each
equation by changing it to slope-intercept form.
3. Complete the table below to guide you in graphing each equation, then answer the
questions that follow.
Equation Slope y-Intercept
1.
2.
a. What is the slope of equation 1? of Equation 2?
b. What is the y-intercept of equation 1? of Equation 2?
c. What can you observe with the slope and y-intercept of equations 1 and 2?
Lesson 2: Solution of a System of Linear Equations in
Two Variables
Warm Up!
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d. What relationship can you infer regarding the slope, y-intercept, and the graph
of the system?
Below are example of pairs of equations that may be used for the activity.
a. and –
b. and
c. and
One way to determine the graph of a system of linear equations is to compare the linear
equations with each other in terms of their slopes and y-intercepts. To do this, as you may
have observed in Warm Up!, it is best to express both linear equations in the slope-
intercept form , where is the slope and is the y-intercept, and check
whether the slopes and y-intercept follow the conditions below;
If the slopes of the two lines are equal, then the graph of the system consists of
either as a set of parallel lines (not equal y-intercepts) or overlapping lines (equal y-
intercepts).
If the slopes are not equal, then the graph of the system consists of a set of
intersecting lines.
Systems of linear equations are also categorized according to the type of solutions.
Learn about It!
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A consistent system can further be identified as either dependent or independent.
All solutions to one of the equations in a dependent system are also solutions to the other
equation.
Recall that the graph of a linear equation in the form is a line, hence, two
such equations can be graphed with two lines on the same rectangular or Cartesian
coordinate plane.
The three types of solutions also have varying graphs.
A consistent-dependent system has a graph that consists of overlapping lines.
Definition 2.4: An independent system has only one
solution.
Definition 2.3: A dependent system has infinitely many
solutions.
Definition 2.2: A system of linear equations is said to be
inconsistent if it has no solution.
Definition 2.1: A system of linear equations is said to be
consistent if it has either one unique or more
than one (infinitely many) solution(s).
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A consistent-independent system has a graph that consists of intersecting lines.
The graph of an inconsistent system consists of parallel lines.
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Example 1: Given the system
, what must be the value of for the system
to be inconsistent?
Solution: An inconsistent system is a system that has no solution because the graph of
the system are parallel lines. Since the graphs are parallel lines, therefore the
slopes must be equal and the y-intercepts must be unequal. Since the given
system already has unequal y-intercepts, must be unequal to .
Try It Yourself!
Given the system
, what must be the value of and for the system to
be consistent and independent?
Example 2: Identify if the system of linear equations
will produce
intersecting, parallel or overlapping lines.
Step 1: Express each equation in slope-intercept form .
Equation 1:
Let’s Practice!
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Equation 2:
Step 2: Find the slope and the y-intercept of each equation.
Equation 1:
;
Equation 2:
;
Step 3: Compare the slopes and the y-intercepts.
Since the slopes are equal, the graph of the system will form either parallel
or overlapping lines. But since the y-intercepts are equal too, the graph
consists of overlapping lines.
Thus, you may conclude that the system of linear equations
has infinitely many solutions, and its graph consists of
overlapping lines.
Try It Yourself!
Identify if the system of linear equations
will produce intersecting,
parallel or overlapping lines.
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Example 3: Identify if the given system of linear equations
will produce
intersecting, parallel, or overlapping lines.
Solution:
Step 1: Express each equation in slope-intercept form.
Equation 1:
Equation 2:
Step 2: Find the slope and the y-intercept of each equation.
Equation 1:
;
Equation 2:
;
Step 3: Compare the slopes and the y-intercepts.
Since the slopes are not equal, this system will form intersecting lines.
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Therefore, we may conclude that the system of linear equations
has a graph consisting of intersecting lines, and has only
one solution.
Try It Yourself!
Identify if the given system of linear equations
will produce
intersecting, parallel, or overlapping lines.
Real-World Problems
Example 4: Joanna is asked to look for two unknown numbers. The
sum of the two numbers is 87. Twice the smaller number
less the larger number is 18. Which among the two pairs
below can be the two unknown numbers?
a. 43 and 44 b. 35 and 52
Solution:
Step 1: Represent the smaller number as and the larger number as .
Let be the smaller number.
be the larger number.
Step 2: Represent the system.
The sum of the two numbers is 87
Twice the larger number less the smaller number is 18.
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Thus, the system would be
Step 3: Since both pairs of numbers add up to 87, we need to verify if difference of
twice the smaller less the larger is 18.
Given 43 and 44.
Given 35 and 52.
Therefore, 35 and 52 are solutions of the system of equations.
Try It Yourself!
Can 32 and 55 be a solution for the two unknown numbers in
Example 4? Justify your answer.
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1. Identify if the given system of linear equations will produce intersecting, parallel, or
overlapping lines.
a.
b.
c.
2. Identify the type of system for each of the given systems in Number 1 as either
consistent-independent, consistent-dependent, or inconsistent.
3. Janice is thinking of two integers. The smaller integer is and the greater integer is
. When she gets the value of , she obtains . What do you think will
be equal to?
Check Your Understanding!
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If I Were, Then U Are
Materials Needed: fishbowl, pen, paper
Instructions:
1. This activity may be done by pair or by group.
2. Your teacher has prepared a fishbowl with rolled small pieces of papers.
3. A member of each group will then pick a piece of paper inside the fishbowl and
then will move at the back of the classroom.
4. You will then be given time to read the question in each piece of paper.
5. Your teacher will then give a signal for you to solve and answer the question on the
board.
6. When you write your solution on the board, you have to write the given and write
your answer in the form “If I were ____, then U are _____.”
7. Your teacher will then verify your answers.
8. The group of the player who gives the right answer first gets a point.
9. The group with the most points wins.
Example of questions:
Given: If I were 4, then you are ______.
Given: If I were 3, then you are ______.
Given: If I were -1, then you are ______.
Lesson 3: Solving Systems of Linear Equations in Two
Variables: Substitution
Warm Up!
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One method to solve a system of linear equations is the substitution method.
In Warm Up!, the substitution method was used to find the value of , however, to solve a
system of linear equation using the substitution method, one of the equations is to be
manipulated to be expressed in one variable in terms of the other variable. This derived
value would then be substituted to the second equation, forming a new equation with
only one variable.
Consider the following system of linear equations:
What values of and satisfy the system?
To solve for and by substitution, the following steps may be helpful:
Step 1: Manipulate one equation so that one variable is in terms of the other.
Note: For convenience, choose an equation with a single variable or one with a
variable whose coefficient is 1.
Solve for in terms of .
Step 2: Substitute the derived expression into the second equation.
Learn about It!
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Step 3: Solve the equation.
Step 4: Substitute the derived value in Step 3 into the equation in Step 1.
Thus, the final answers are and . This solution can also be written as the
point .
Example 1: Solve the following system of linear equations using substitution.
Let’s Practice!
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Solution:
Step 1: Since Equation 2 is already expressed in terms of , we can quickly substitute
it into Equation 1.
Step 2: Substitute the expression in Step 1 into the first equation.
Step 3: Solve the equation.
Step 4: Substitute the derived value in Step 3 into the equation in Step 1.
Thus, the solutions are and . This solution can also be written as
the ordered pair .
Try It Yourself!
Solve the following system of linear equations using substitution.
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Example 2: Solve the following system of linear equations using substitution.
Solution:
Step 1: Manipulate one equation so that one variable is in terms of the other.
Note: For convenience, choose an equation with a single variable or one with a
variable whose coefficient is 1.
The first equation has a with a coefficient of 1.
Solve for in terms of .
Step 2: Substitute the derived expression into the second equation.
Step 3: Solve the resulting equation.
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Step 4: Substitute the derived value in Step 3 into the equation in Step 1.
Thus, the solutions are and . This solution can also be written as
the ordered pair .
Try It Yourself!
Solve the following system of linear equations using substitution.
Example 3: Solve the system of linear equations.
Solution:
Step 1: Manipulate one equation so that one variable is in terms of the other.
Solve for in terms of .
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Step 2: Substitute the derived expression into the second equation.
Step 3: Solve the equation.
Step 4: Substitute the derived value in Step 3 into the equation in Step 1.
Thus, the solutions are and . This solution can also be written as
the ordered pair .
Try It Yourself!
Solve the following system of linear equations using substitution.
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Real-World Problems
Example 4: Go Green! purchased 51 ornamental trees to plant in a
park. They bought a number of small and large trees. If
the number of the small trees is twice the number of the
large trees, how many trees of each size did they plant?
Solution:
Step 1: Represent the number of small trees as and the number of large trees as .
Let be the large trees
be the small trees
Step 2: Represent the system.
There are 51 trees in all.
The number of small trees is twice the number of large trees.
Thus, the system would be
Step 3: Since equation 2 is already expressed in terms of , we can substitute the
value of to the first equation.
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Step 4: Substitute the derived value in Step 3, to
Therefore, there are 17 large trees and 34 small trees.
Try It Yourself!
The length of the top of a rectangular table is 5 more than twice its
width. Its perimeter is 70 cm. What is its area?
1. Solve the following systems of linear equations by substitution.
a.
b.
2. Analyze and solve the following problem:
There were 48 students who auditioned for a school’s Glee Club. There were three
times as many female students who auditioned as there were males. How many
males and how many females auditioned for the school’s Glee Club?
Check Your Understanding!
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Neutral Pairs
Materials Needed: paper and pen, black and white cartolinas
Instructions:
1. This activity may be done individually or in pairs.
2. Create algebra tiles using the black and white cartolinas.
3. Label the tiles accordingly: x-tiles are the small rectangular tiles, y-tiles are the big
rectangular tiles while the small square tiles are 1 unit each.
4. Given and , model the equations using the algebra tiles.
5. Combine the two equation models.
6. Remove the neutral pairs and write the new equation.
7. Answer the following questions:
a. Which variable was removed?
b. What is the resulting equation when the variable was removed?
c. Can you solve the system using these method? Justify your answer.
8. Present your findings in class.
Lesson 4: Solving Systems of Linear Equations in Two
Variables: Elimination
Warm Up!
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Another way to solve systems of linear equations algebraically, besides substitution, is
through elimination of variables. In elimination, either or is removed or eliminated
leading to an equation in one variable. The variable is retained when is eliminated or
removed and vice-versa. Using the algebra tiles in the Warm Up! activity is one way of
showing the elimination of variables.
Algebraically, elimination is a technique for solving systems of linear equations which
involves cancelling, or eliminating, one variable by adding or subtracting the two
equations.
A key step in the process is multiplying either or both of the equations by a constant such
that the coefficients for either or have the same absolute value but opposite signs.
To understand it better, consider the following system of linear equations:
Which variable has coefficients with the same absolute value?
The coefficients of the variable have the same absolute value; that is, 1 and –1. Hence, to
eliminate , the two equations must be added.
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Solving for , we have
The derived value for can now be substituted into any of the given equations to find .
Therefore, the solution to the system is .
Example 1: Solve the following system using elimination:
Solution:
Step 1: Choose a variable to eliminate.
By inspection, it is easier to eliminate instead of since the coefficients of
in both equations have the same absolute value and
Step 2: Add the two equations to eliminate the chosen variable.
Let’s Practice!
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Step 3: Solve the resulting linear equation.
Step 4: Find the value of the second variable.
Substitute the value of the variable you found to any of the original
equations.
Thus, the solution of the system is and . This solution may also
be expressed as the ordered pair .
Try It Yourself!
Solve the following system of linear equations using elimination:
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Example 2: Solve the following system using elimination:
Solution:
Step 1: Choose a variable to eliminate.
Let us say, we choose to eliminate .
Step 2: Manipulate the equations so that the coefficients of the chosen variable in
both equations have the same absolute value but different signs.
Multiply the first equation by because from the second equation has a
coefficient of 2.
Step 3: Add the two equations to eliminate the chosen variable.
Step 4: Solve the resulting linear equation.
Step 5: Find the value of the second variable.
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Substitute the value of the variable you found to any of the original
equations.
Thus, the solution of the system is and . This may be
expressed as the ordered pair
Try It Yourself!
Solve the following system of linear equations using elimination:
Example 3: Solve the following system using elimination:
Solution:
Step 1: Choose a variable to eliminate.
It can be seen that it is easy to find a multiplier for in the second equation
to have the same absolute value as the coefficient of in the first equation.
Thus, we choose to eliminate .
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Step 2: Manipulate the equations so that the coefficients of the chosen variable in
both equations have the same absolute value but of opposite signs.
Multiply the second equation by 10.
Step 3: Add the two equations to eliminate the chosen variable.
Step 4: Solve the resulting linear equation.
Step 5: Find the value of the other variable.
Substitute the value of the variable you found to any of the original
equations.
–
Thus, the solution of the system is and . This may be expressed
as the ordered pair .
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Try It Yourself!
Solve the following system of linear equations using elimination:
Real-World Problems
Example 4: Soybean meal is 16% protein while cornmeal is 9% protein. How
many pounds of each should be mixed together to get a 350 lb
mixture that is 12% protein?
Solution:
Step 1: Let number of pounds of soybean meal.
number of pounds of cornmeal.
Step 2: List important information in a table.
Soybean Cornmeal Mixture
Amount of meal
to be mixed
Percent of
protein
Amount of
protein in
mixture
Note: 42 was obtained by getting 12% of 350.
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Step 3: Represent the system.
Note: The system may be transformed into the following by multiplying
Equation 2 by 100 for easier computation:
Step 4: Start solving the system by multiplying the first equation by to eliminate
the variable .
Step 5: Add the two equations to eliminate the chosen variable.
Step 4: Solve for .
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Thus, the mixture should contain 150 pounds of soybean meal and 200
pounds of cornmeal.
Try It Yourself!
Solution A is 2% alcohol and solution B is 6% alcohol. A drugstore
owner wants to mix the two solutions to get a 60-liter solution that is
3.2% alcohol. How many liters of each solution should the owner use?
1. Solve the following systems of linear equations by elimination:
a.
b.
2. Analyze and solve the following problem:
A jet can travel 1200 kilometers in 4 hours with the wind. The return trip against the
wind takes 6 hours. Find the rate of the jet in still air and the rate of the wind.
Check Your Understanding!
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It’s Graphing Time!
Materials Needed: smartphone or tablet with a graphing application, pen, paper,
graphing paper, ruler (optional)
Instructions:
1. This activity may be done in groups of 3.
2. Using the graphing application in your smartphone or tablet, input the linear
equations listed. You may seek the assistance of your teacher in doing so if you are
not familiar with the application. If the use of electronic gadgets is not allowed, you
may opt to graph the equations in a large piece of graphing paper.
3. After graphing the equations, observe the relationship between the graphs of the
given equations then complete the table that follows.
Equations:
Equation 1:
Equation 2:
Equation 3:
Equation 4:
Equation 5:
Equation 6:
Lesson 5: Solving Systems of Linear Equations in Two
Variables: Graphing
Warm Up!
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Equations What can you
say about
their graphs?
Is there a
common
point?
What type of
system is
given?
How many
solutions are
there?
Equations 1
and 2
Equations 3
and 4
Equations 5
and 6
Aside from algebraic methods like substitution and elimination, another way of finding
the solution of a system of linear equations in two variables is by graphing or by the
graphical method. Solving a system of linear equations in two variables by graphing will
result in three cases as observed in the Warm Up! activity.
1. If the graphs of the two equations form intersecting lines (consistent and
independent), then the system has a unique solution.
Learn about It!
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2. If the two equations represent the same line; that is, the graphs form overlapping
lines (dependent), then the system has infinitely many solutions.
3. If the graphs of the two equations form parallel lines (inconsistent), then the
system has no solution.
In graphing an equation, it is best to rewrite it first into the slope-intercept form
, where represents the slope, and is the y-intercept.
The following systems of linear equations are graphed to determine the solution.
1. Unique solution
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The point of intersection is (−1, 3). This point represents the solution to the system.
2. Infinitely many solutions
This is a case of overlapping lines. Thus, the system has infinitely many solutions.
This means all the points on the line are solutions to the system.
3. No solution
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The graph shows two parallel lines. Note that they do not intersect. Thus, this
system has no solution.
Example 1: Find the solution set of the system of linear equations shown below by
graphing using their x and y intercepts.
Solution:
Step 1: Determine the x and y-intercepts of each equation in the given system.
For
If , then The y-intercept is .
Let’s Practice!
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If , then The x-intercept is .
For
If , then The y-intercept is .
If , then The x-intercept is .
Step 2: Graph each line using the intercepts.
Plot the intercepts and draw a straight line through both points.
Step 3: Identify the solution based on the graph.
The graph is a pair parallel lines. Thus, the system has is inconsistent. It has
no solution.
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Try It Yourself!
Find the solution set of the system of linear equations shown below by graphing.
Example 2: Find the solution set of the system of linear equations shown below by
graphing.
Solution: This time, we shall use the slope-intercept form in graphing.
Step 1: Graph the first linear equation.
Graph using the slope and the y-intercept.
The slope is −2, and the y-intercept is 4, or (0, 4).
Thus, from the point (0, 4), go down 2 units, then count 1 unit to the right to
get the second point (1, 2). Connect the two points to form a line.
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Step 2: Graph the second linear equation.
Graph using intercepts.
Find the x-intercept:
The x-intercept is .
Find the y-intercept:
The y-intercept is .
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Plot the intercepts and draw a straight line through both points.
Step 3: Identify the solution based on the graph.
The graph is a set of parallel lines. Thus, the system has no solution.
Try It Yourself!
Find the solution set of the system of linear equations shown below by graphing
using their x- and y-intercepts.
Example 3: Find the solution set of the system of linear equations shown below by
graphing.
Solution:
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Step 1: Graph the first linear equation.
Graph using the slope and the y-intercept. Rewrite the equation in slope-
intercept form.
The slope is −2, and the y-intercept is −4, or
Step 2: Graph the second linear equation.
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Graph using the slope and the y-intercept. Rewrite the equation in slope-
intercept form.
The slope is −3, and the y-intercept is 1, or .
Step 3: Identify the solution based on the graph.
The graph is a set of lines intersecting at Thus, the system has a
unique solution: and .
Try It Yourself!
Find the solution set of the system of linear equations shown below by graphing.
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Real-World Problems
Example 4: A total of ₱25 000 is invested in a bank in two funds
paying 6% and 8% annual interest. The combined
annual interest is ₱1800. How much of the ₱25 000 is
invested in each fund?
Solution:
Step 1: Let x be the amount invested in 6% annual interest.
y be the amount invested in 8% annual interest.
Step 2: List important information in a table.
Total Amount
Annual Interest
Total Interest Earned
Step 3: Represent the system
Note: The system may be transformed into the following by multiplying
Equation 2 by 100 for easier computation:
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Step 4: Graph the system.
Equation 1 has intercepts at (25000, 0) and (0, 25000).
Equation 2 has intercepts at (30000, 0) and (0, 22500).
The two lines intersect at (10000, 15000), thus, the amount invested at 6% is
₱10 000 while the amount invested at 8% interest rate is ₱15 000.
Try It Yourself!
A total of ₱50 000 is invested in two funds paying 5% and 6% annual interest. The
annual interest is ₱2 800. Find how much of the ₱50 000 pesos is invested in each?
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1. Find the solution set of the system of linear equations shown below by graphing.
a.
b.
2. On a municipal hospital, about 51 babies are born every day. Of these babies, there
are about twice as many girls as there are boys. How many boys and girls are born
each day?
1. If a system of linear equations may be solved through graphing, why do you think
there is a need to know the algebraic methods (substitution and elimination)? On
what instances do you think is it more convenient to use the substitution method
over elimination method, and vice versa?
2. Write an equation which can be paired to the equation to form a
system with a solution of .
Challenge Yourself!
Check Your Understanding!
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3. Find the system of linear equations with the following graph.
This activity will showcase your learning in this unit. You will assume the role of a travel
specialist/advisor working in a travel company.
The company is launching their new travel packages for various types of customers. As a
specialist, you are tasked to provide detailed travel packages and plans in the form of
brochures highlighting the different travel destinations in Luzon. Specifically, you are to
provide camping and hiking destinations as well as computations for camping materials
and consumables. The materials include tents, ropes, cooking utensils, as well as other
items you may think are necessary. Food packages are also provided for the
campers/travelers in an optional basis.
You may refer to the given format below as guide for your brochure.
Performance Task
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I. Travel Destination
Location
Short Description
Images
II. Basic Travel Packages for:
Single (₱2000 per pax)
Couples (₱1500 per pax)
Family (₱1000 per pax)
Friends/Barkada (₱1000 per pax)
Team Building/Company (₱750 per pax)
III. Materials and Food
You are going to present the brochure and plan to the head of your travel agency for
approval. Once approved, the brochure will then be printed for circulation and marketing
purposes. As part of the planning you must provide data for the following inquiries below
prior to the presentation.
1. List of all camping materials needed. Specify the quantity for each as well as the
price. Keep in mind that the materials should be appropriate based on the travel
package and the needs of the travelers (e.g. adults, kids, male or female)
2. List of all ingredients needed for the menu for the food packages. Specify the
quantities needed and the unit price for each ingredient. Keep in mind that the
ingredients should be appropriate based on the travel package and the needs of
the travelers (e.g. adults, kids, male or female)
3. The data can be presented in tabular form. You may refer to the following table as
your guide.
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TOUR PACKAGE MATERIALS FOOD INGREDIENTS
NAME QUANTITY COST NAME QUANTITY COST
4. Use the data from the table to formulate at least 5 problems involving systems of
linear inequalities in two variables then solve each problem.
The brochure and plan will be evaluated according to the following: design and
presentation, accuracy, usefulness, and mathematical justification.
Performance Task Rubric
Criteria
Below
Expectation
(0–49%)
Needs
Improvement
(50–74%)
Successful
Performance (75–
99%)
Exemplary
Performance
(99+%)
Design and
Presentation
The design and
presentation is
poor.
The design and
presentation is
somewhat
informative.
The design and
presentation is
informative and
flawless.
The design and
presentation is
very informative
and flawlessly
done. It is also
easy to
understand.
Accuracy
The information
given are
erroneous and do
not show wise use
The information
given are
erroneous and
show some use of
The information
given are accurate
and shows use of
the concepts of
The information
given are accurate
and show a wise
use of the
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of the concepts of
systems of linear
equations in two
variables.
the concepts of
systems of linear
equations in two
variables.
systems of linear
equations in two
variables.
concepts of
systems of linear
equations in two
variables.
Usefulness
The brochure is
not useful in
understanding
the information
given.
The brochure is
somewhat useful
in understanding
the information
given.
The brochure is
well-crafted and
useful for
understanding the
information given.
It showcases the
proper layout.
The brochure is
well-crafted and
useful for
understanding the
information given.
It showcases the
proper layout and
is accurately done.
Mathematical
Justification
Justification is
ambiguous. Only
few concepts of
systems of linear
equations in two
variables are
applied.
Justification are
not so clear. Some
ideas are not
connected to each
other. Not all
concepts of
systems of linear
equations in two
variables are
applied.
Justification is
clear and
informatively
delivered.
Appropriate
concepts learned
on systems of
linear equations in
two variables are
applied.
Justification is
logically clear,
informative, and
professionally
delivered. The
concepts learned
on systems of
linear equations in
two variables are
applied.
Systems of Linear Equations in Two
Variables
Solving Systems of Linear Equations in
Two Variables
Algebraic Method:
Elimination and
Substitution
Graphical Method
Wrap-up
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Key Terms/Formulas
Concept Relevant Information
System of Linear
Equations
A system of linear equations involves two or more linear
equations.
The solution of a system of linear equations in two
variables involves a value of x and y that satisfies both
equations.
Graphically, the solution is the point of intersection of the
two lines.
Solutions of System of
Linear Equations
A consistent system of two linear equations has one or
more solutions. A consistent system can be either
dependent or independent.
A dependent system has infinitely many solutions. All
solutions for one of the equations are also solutions to the
other equation. Its graph consists of two overlapping lines.
An independent system has one solution. Its graph
consists of two intersecting lines.
An inconsistent system has no solution. Its graph consists
of two parallel lines.
Solving Systems of
Linear Equations by
Substitution
The substitution method is a technique in solving system
of linear equations that expresses one variable in terms of
the other variable. This derived expression would then
be substituted to the second equation, forming a new
equation with only one variable.
Solving Systems of
Linear Equations by
Elimination
Elimination is a technique for solving systems of linear
equations that involves cancelling, or eliminating, one
variable by adding or subtracting the two equations.
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Solving Systems of
Linear Equations by
Graphing
Graphing is one method to solve a system of linear
equations in two variables.
A system with a unique solution is represented by the
point of intersection of two lines.
A system with no solution is represented by two parallel
lines.
A system with infinitely many solutions is represented by
overlapping lines.
Lesson 1
1.
, or answers may vary.
2. Yes
3. No
4.
Lesson 2
1. and can be any number as long as
2. intersecting lines
3. overlapping lines
4. No, because the pair of numbers given cannot satisfy the two equations.
Lesson 3
1.
2.
3.
4. 250 sq. units
Key to Let’s Practice!
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Lesson 4
1.
2.
3.
4. 42 liters of solution A and 18 liters of solution B
Lesson 5
1. The graph is a set of parallel lines. Thus, the system has no solution.
2. The graph is a set of lines intersecting at . Thus, the system has a unique
solution: and .
3. The system has infinitely many solutions since the graph of the lines are
overlapping.
4. The amount invested at 5% interest is ₱20,000 while the amount invested at 6%
interest is ₱30,000.
Abuzo, Emmanuel P., et al. Mathematics Learners’ Module Grade 8. Book Media Press Inc.,
2013
Baron, Lorraine, et al. Math Makes Sense 8. Canada: Pearson Education, 2008.
Maths Is Fun. “Algebra”. Accessed February 28, 2018.
http://www.mathsisfun.com/algebra/systems-linear-equations.html
McGraw-Hill Education. Glencoe Math Volume 1. McGraw-Hill Professional, 2013.
Oronce, Orlando A., et. al. E-Math Intermediate Algebra. Philippines: Rex Bookstore, Inc.
2007
References