daily warm-up 7.1 daily warm-up is the ordered pair a solution of the equation 2x – 3y = 5? 1.(1,...

Daily Warm-Up 7.1

Daily Warm-Up

Is the Ordered Pair a Solution of the equation 2x – 3y = 5?

1. (1, 0)

2. (-1, 1)

3. (1, -1)

Tonight’s HomeworkGraphing Systems Worksheet #1-10 + PREVIEW L 7.2

5.6

7.1 Solving Linear Systems by Graphing

Objective: Solve a system of linear equations by graphing

What is a System of Linear Equations?

A system of linear equations is simply two or more linear equations using the same variables.

We will only be dealing with systems of two equations using two variables, x and y.

If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x , y) where x and y make both equations true at the same time.

We will be working with the graphs of linear systems and how to find their solutions graphically.

How to Use Graphs to Solve Linear Systems

x

yConsider the following system:

x – y = –1

x + 2y = 5Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line.

We can also see that any of these points will make the second equation true.

However, there is ONE coordinate that makes both true at the same time…

(1 , 2)

The point where they intersect makes both equations true at the same time.

x – y = –1

x + 2y = 5

How to Use Graphs to Solve Linear Systems

x

yConsider the following system:

(1 , 2)

We must ALWAYS verify that your coordinates actually satisfy both equations.

To do this, we substitute the coordinate (1 , 2) into both equations.

x – y = –1

(1) – (2) = –1 Since (1 , 2) makes both equations true, then (1 , 2) is the solution to the system of linear equations.

x + 2y = 5

(1) + 2(2) =

1 + 4 = 5

Graphing to Solve a Linear System

While there are many different ways to graph these equations, we will be using the slope – intercept form.

To put the equations in slope intercept form, we must solve both equations for y.

Start with 3x + 6y = 15

Subtracting 3x from both sides yields

6y = –3x + 15

Dividing everything by 6 gives us…51

2 2y x=- +

Similarly, we can add 2x to both sides and then divide everything by 3 in the second equation to get

23 1y x= -

Now, we must graph these two equations

Solve the following system by graphing:

3x + 6y = 15

–2x + 3y = –3


512 2

23 1

y x

y x

=- +

= -

Solve the following system by graphing:

3x + 6y = 15

–2x + 3y = –3

Using the slope intercept forms of these equations, we can graph them carefully on graph paper.

x

y

Start at the y – intercept, then use the slope.Label the solution!

(3 , 1)

Lastly, we need to verify our solution is correct, by substituting (3 , 1).

Since and , then our solution is correct!( ) ( )3 3 6 1 15+ = ( ) ( )2 3 3 1 3- + =-


Let's summarize! There are 4 steps to solving a linear system using a graph.

Step 1: Put both equations in slope – intercept form

Step 2: Graph both equations on the same coordinate plane

Step 3: Estimate where the graphs intersect.

Step 4: Check to make sure your solution makes both equations true.

Solve both equations for y, so that each equation looks like

y = mx + b.

Use the slope and y – intercept for each equation in step 1. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution!

Substitute the x and y values into both equations to verify the point is a solution to both equations.

x

y

LABEL the solution!




Step 3: Estimate where the graphs intersect. LABEL the solution!


Let's do ONE more…Solve the following system of equations by graphing.

2x + 2y = 3

x – 4y = –1

32y x=- +

1 14 4y x= +

( ) ( )122 1 2 2 1 3+ = + =

( )121 4 1 2 1- = - =-

( )121,

x

y




Step 3: Estimate where the graphs intersect. LABEL the solution!


It’s your turn! … Solve the following system of equations by graphing.

3x + 2y = 2

x = –1

Summary

1. When you are asked to solve the linear equation system by graphing, you only need to follow the four-step:

Step 1: Put both equations in slope – intercept form: y = mx + b

Step 2: Graph both equations on the same coordinate plane: use the y-intercept as the first point and then use slope to create the second point.

Step 3: Estimate where the graphs intersect, and also label the intersection which is the solution of the linear equation system.

Step 4: Check to make sure your solution makes both equations true because the graphing may contain some error. Substitute the x and y values into both equations to verify the point is a solution to both equations.

2. Don’t forget the special line!!!

Guided Practice – L 7.1 DHQ

DAILY HOMEWORK QUIZPAGE: 401 in Textbook

Numbers: 8-10 on graph paper!

* ALL ANSWERS ARE ORDERED PAIRS!

Tonight’s Homework:Graphing Systems Worksheet #1-10 + PREVIEW L 7.2

Daily Warm-Up 7.25.6

x

y

Find the Solution to the System of Equations. Write your solution as an ordered pair (x, y).

2x – 3y = 9

X = -3

Tonight’s Homework:Substitution WS #1-10 + PREVIEW L 7.3

7.2 Solving Systems of Equations using SubstitutionObjective: Use substitution method to solve a linear system

Solving Systems of Equations using Substitution

Steps:

1. Solve one equation for one variable (y = ; x = ; a =)

2. Substitute the expression from step one into the other equation.

3. Simplify and solve the equation.

4. Substitute back into either original equation to find

the value of the other variable.

5. Check the solution in both equations of the system.

Example #1: y = 4x3x + y = -21

Step 1: Solve one equation for one variable.

y = 4x (This equation is already solved for y.)

Step 2: Substitute the expression from step one into the other equation.

3x + y = -21

3x + 4x = -21

Step 3: Simplify and solve the equation.

7x = -21

x = -3

y = 4x3x + y = -21

Step 4: Substitute back into either original equation to find the value of the other variable.

y = 4x y = 4(-3) = -12

or 3x + y = -21 3(-3) + y = -21 -9 + y = -21 y = -12

Solution to the system is (-3, -12).

This substitution is easier

y = 4x3x + y = -21

Step 5: Check the solution in both equations.

y = 4x

-12 = 4(-3)

-12 = -12

3x + y = -21

3(-3) + (-12) = -21

-9 + (-12) = -21

-21= -21

Solution to the system is (-3,-12).

Example #2: x + y = 10 5x – y = 2

Step 1: Solve one equation for one variable.

x + y = 10

y = –x +10Step 2: Substitute the expression from step one into

the other equation.

5x – y = 2

5x – (–x +10) = 2

x + y = 10 5x – y = 2

5x – (–x + 10) = 2

5x + x – 10 = 2

6x – 10 = 2

6x = 12

x = 2

Step 3: Simplify and solve the equation.

x + y = 10 5x – y = 2

Step 4: Substitute back into either original equation to find the value of the other variable.

x + y = 102 + y = 10 y = 8

Solution to the system is (2,8).

x + y = 10 5x – y = 2

Step 5: Check the solution in both equations.

x + y =10

2 + 8 =10

10 =10

5x – y = 2

5(2) - (8) = 2

10 – 8 = 2

2 = 2

Solution to the system is (2, 8).

You try Exercise 1

Solve the system using the substitution method.y = –63 – 7x3x + 7y = –73

A) (8, –8) B) (–8, –7) C) (–7, –8) D) (–7, 9)

YOU TRY: Solve by substitution:

2 3 7

2 5

a b

a b

Summary

1. When solving the linear systems, it is very important to remember the 5 steps and follow them in the order.

2.When in a certain step the equation reduces to a contradiction, you could conclude that the system is inconsistent and has no solution.

3.When in a certain step the equation reduces to an identity, you could conclude that the system is dependent and has many solutions.


PAGE: 408 in Textbook

# 7• ANSWERS ARE ORDERED PAIRS!• MUST SHOW WORK

Tonight’s Homework:

Substitution WS #1-8, 10 (SKIP #9)

+ PREVIEW L 7.3

Daily Warm-Up 7.3

x

y

Find the Solution to the System of Equations using 2 methods, GRAPHING and SUBSTITUTION.Write your solution as an ordered pair (x, y).

y = 1

-2x + y = - 1

Tonight’s Homework:Elimination WS #1-10 + Study for QUIZ (L 1-3) – QUIZ IS THURSDAY (1/27)

7.3 Solving Linear Systems by Linear Combinations

(Elimination) Method

Objective: Use linear combinations to solve a system of linear equations.

What is a System?

A system of linear equations is:

a. A set of parabolas

b. A set of two or more lines

c. A stereo component

Linear Combinations (Elimination) Method

Linear Combinations (Elimination) method is used when it appears easy to eliminate one variable from the system through transformation. Remember that linear transformations do not change the solutions of a system.

Linear Combinations (Elimination) Method

Step 1 Make sure the variables are lined up properly by their names

-2x + y = 4-6x + y = 0

Step 2 Make the coefficients of one of variables opposites (Multiply one or both equations by appropriate numbers so that the coefficients in one variable are opposites). Correctly applying the distributive property is very important!!!

In this question, notice that the y coefficients are 1, therefore we can multiply either equation by -1 and add the system, thus eliminating the y variable.

Linear Combination (Elimination) Method

Step 2 Let’s transform the second equation:-1(-6x + y = 0) 6x – y = 0

Step 3 Add the new two equations up: -2x + y = 4 + 6x – y = 0 4x = 4


Step 4 Solving the resulted one variable equation for x

4x = 4yields x = 1.

Step 5 Back substitute to one of the equation in the system to find the value for the other variable. Once we have the x value, we can plug it into either of our original equations and solve for y:

-2x + y = 4 -6x + y = 0


Step 5 Plugging x = 1 into the first equation yields:

-2 (1) + y = 4-2 + y = 4, y = 6

Step 6 Check the answer. -2(1) + 6 = 4-6(1) + 6 = 0

So our solution is (1, 6).


Let’s Try One More Together! Use the elimination method to solve the system:

x – 5y = –23x + 2y = 11

The first step is:a. Add the equations togetherb. Transform the first equation by multiplying it

by –3c. I’m not sure. I need to review the eliminatio

n method.


Yes, notice that if we multiply the first equation by -3, we obtain additive inverses for the x coefficient. Our system is now:

-3x + 15y = 6 3x + 2y = 11

The next step is:a. Add the two equationsb. Solve the second equation for yc. I’m not sure. I need to review.

Yes, by adding the two equations we get: – 3x + 15y = 6

+ 3x + 2y = 11 17y = 17

Solving for y, we get y = 1. The next step is:a. Plug y = 1 into either equation and solve fo

r xb. Plug x = 1 into either equation and solve fo

r y

c. I’m not sure. I need to review.


We found that y=1, not x=1. So we must plug y=1 into either equation to solve for x not for y.


Yes, now we can plug y = 1 into either of the original equations, or the transformed equation. Let’s choose the first original equation:

x – 5y = –2 x – 5(1) = –2 x – 5 = –2 x = 3

So our solution is (3, 1), which must be the only solution to a system of two lines.

Next, we must check the solved order pair is the solution to the system. (omitted here)


You TrySolve the system using elimination method:

4x + 8y = 20

-4x + 2y = -30

The solution is:

a. (1, 7)

b. (3, -2)

c. (7, -1)

d. (-7, -1)

You TrySolve the system using elimination method:

2x + 5y = 7

3x + y = -9

The solution is:

a. (3, -4)

b. (-4, -3)

c. (-4, 3)

d. (-3, -4)

Summary1. The key concepts of the linear combinations

(elimination) method are to(a) variables are lined up properly(b) find out if there are any opposite

coefficients for the same variable. If yes, directly add two equations up and eliminate one variable. If not,

(c) multiply (or divide) some number(s) to equation(s) so that two opposite coefficients for the same variable show up. Then follow (b).

2. Some problems can be applied both substitution and linear combination (elimination) methods. However, it is better to use linear combination (elimination) method than to use the substitution method to some other problems. Choosing an appropriate method is important.



Number: 4• ANSWERS ARE ORDERED PAIRS!

Tonight’s Homework:

Elimination WS #1-10 + Study for QUIZ (L 1-3) – QUIZ IS THURSDAY (1/27)

936 yx

Rewrite the equations to Slope-Intercept Form and Graph.( y = mx + b )

Daily Warm-Up 7.5

Tonight’s Homework:Textbook Pages 429-430 #12-29 (USE ANY METHOD)

7.5 Special Types of Liner Systems

Objectives

1) Identify linear system algebraically as having one solution, no solution, and infinite many solutions.

2) Identify linear system geometrically as having one solution, no solution, and infinite many solutions.

Slope-Intercept Form

Example 1

(2) 72

(1) 523

yx

yxSolve the linear equation system

Solution 1

Substitution

x = –2y + 7From (2)

3(–2y + 7) – 2y = 5To (1)

–6y + 21 – 2y = 5

–8y = –16

y = 2then

x = –2(2) + 7 = 3

(2) 7)2(2 3

(1) 5)2(2)3(3

The solution is (3, 2).

Special Types of Linear Systems

Example 1

(2) 72

(1) 523

yx

yxSolve the linear equation system

Solution 2

Graphing

523 yx 532 xy

2

5

2

3

xy2

5

2

3 xy

2

3

run

risem

72 yx 72 xy

2

7

2

1

xy

2

7

2

1

xy

2

1

run

risem

(3 , 2)


NUMBER OF SOLUTIONS OF A LINEAR SYSTEM

y

x

Lines intersectone solution


A Linear System with No Solution

Show that this linear system has no solution.

2 x y 5 Equation 1

2 x y 1 Equation 2

Because Equation 2 can be revised to y –2 x 1, you can substitute –2 x 1 for y in Equation 1.

2 x y 5

2 x (–2 x 1) 5 Substitute –2 x 1 for y.

1 5 Simplify. False statement!

Once the variables are eliminated, the statement is not true regardlessof the values of x and y. This tells you the system has no solution.

Write Equation 1.

METHOD 1: SUBSTITUTION

A Linear System with No Solution

Show that this linear system has no solution.

Rewrite each equation in slope-intercept form.

5 4 3 2 1 0 1 2 3 4 51

1

2

3

4

5

6

y –2 x 5 Revised Equation 1

Graph the linear system.

y –2 x 1 Revised Equation 2

2 x y 5 Equation 1

2 x y 1 Equation 2

The lines are parallel; they have the same slope but different y-intercepts. Parallel lines never intersect, so the system has no solution.

y 2x 1y 2x 1

y 2x 5y 2x 5

METHOD 2: GRAPHING

• When solving a system of linear equations, if the resulting equation in certain step reduces to an inconsistent equation (contradiction), then the system has no solution and is inconsistent.

• Geometrically speaking, the two equations in the system represent two parallel lines which never intersect.



y

x

Lines are parallelno solution


IDENTIFYING THE NUMBER OF SOLUTIONS

Practice 2

Use the following method to solve the equation system

(1) Substitution (2) Graphing

2x – 4y = -2x – 2y = 0

Practice 2 Solve the linear equation system

Solution 1

Substitution

x = 2yFrom (2)

2(2y) – 4y = –2To (1)

4y – 4y = –2

0 = –2

This is an inconsistent equation

So the system has no solution.


2 4 2 (1)

2 0 (2)

x y

x y


Solution 2

Graphing

2 4 2x y 4 2 2y x

2 2

4 4y x

1

2

risem

run

2 0x y 2 0y x

1 0

2 2y x

1

02

y x

1

2

risem

run


2 4 2 (1)

2 0 (2)

x y

x y

1 1

2 2y x

Show that this linear system has infinitely many solutions.

– 2 x y 3 Equation 1

– 4 x 2y 6 Equation 2METHOD 1: LINEAR COMBINATIONS

You can multiply Equation 1 by –2.

4x – 2y – 6 Multiply Equation 1 by –2.

– 4 x 2y 6 Write Equation 2.

0 0 Add Equations. True statement!

The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This result tells you that the linear system has infinitely many solutions.

A Linear System with Many Solutions

A Linear System with Many Solutions

Show that this linear system has infinitely many solutions.

– 2 x y 3 Equation 1

– 4 x 2y 6 Equation 2

y 2 x 3 Revised Equation 1

y 2 x 3 Revised Equation 2

Rewrite each equation in slope-intercept form.

Graph the linear system.

From these graphs you can see that the equations represent the same line. Any point on the line is a solution.

5 4 3 2 1 0 1 2 3 4 51

1

2

3

4

5

6

METHOD 2: GRAPHING

– 4x 2y 6– 4x 2y 6–2x y 3–2x y 3


• When solving a system of linear equations, if the resulting equation in certain step reduces to an identity equation, then the system has infinite many solutions and is dependent.

• Geometrically speaking, the two equations in the system represent two identical lines which always intersect. Any point on the first line (the solution to the first equation) is also on the second line (the solution to the second equation). The two identical lines have infinitely many intersection.


y

x

Lines coincideinfinitely many

solutions


solutions


(the coordinates of every point on the line)

Practice 3

Use the following method to solve the equation system

(1) Linear Combination (2) Graphing

3x – 4y = -4-6x + 8y = 8


Solution 1

Linear Combination

2(3x – 4y)= 2(– 4)

Multiply 2 to (1)

6x – 8y = –8 (1)

-6x + 8y = 8 (2)

0 = 0 This is an identity. So the equation system has many solutions.


3 4 4 (1)

6 8 8 (2)

x y

x y


Solution 2

Graphing

3 4 4x y 4 3 4y x

3 4

4 4y x

3

4

risem

run

6 8 8x y 8 6 8y x

6 8

8 8y x

31

4y x

3

4

risem

run


3 4 4 (1)

6 8 8 (2)

x y

x y

31

4y x



y

x




y

x


solutions


solutions


(the coordinates of every point on the line)



CONCEPT

SUMMARY

y

x

y

x



y

x

Lines coincideinfinitely many solutions

Independent Inconsistent Dependent

Summary

Similar to the situations when solving the one variable linear equations, solving the linear systems has the similar 3 situations:(a) One solution -- two lines intersect -- the

system is independent(b) No solution -- two lines are parallel -- the

system is inconsistent -- the system contains a contradiction

(c) Many solution -- two lines are coincide -- the system is dependent -- the system contains an identity



Numbers: 6-8

• USE ANY METHOD

• ANSWERS ARE ORDERED PAIRS!

Tonight’s Homework:Textbook Pages 429-430 #12-29 (USE ANY METHOD)

daily warm-up 7.1 daily warm-up is the ordered pair a solution of the equation 2x – 3y = 5? 1.(1,...

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