ch. 8.2-3: solving systems of equations algebraically
TRANSCRIPT
Ch. 8.2-3: Solving Systems of Equations Algebraically
What is a System of Linear Equations?
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 only be dealing with systems of two equations using two variables, x and y.
We will be working with the graphs of linear systems and how to find their solutions graphically.
A system of linear equations is simply two or more linear equations using the same variables.
inconsistentconsistent
dependentindependent
Definitions Consistent: has at least one solution Dependent: has an infinite amount of
solutions
Inconsistent: has no solutions
Independent: has exactly one solution Consistent: has at least one solution
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.
How to Use Graphs to Solve Linear Systems
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
Graphing to Solve a Linear System
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- + =-
Graphing to Solve a Linear System
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.
SubstitutionSubstitution1. Given two equations. Solve one
equation for a variable.2. Plug this expression in for the variable
into the other equation.3. Solve. 4. Plug this value into the first equation
and solve it as well. 5. Write the answer as an ordered pair.
Example: Solve 1. Solve for p2. Plug into other
equation and solve for q.
3. Plug value of q into initial equation.
4. Write the ordered pair (ABC order)
2 3 2
3 17
p q
p q
3 17p q 17 3p q
2 3 2p q 2( 17 3 ) 3 2q q
34 6 3 2q q 9 36q
4q 3 17p q 3(4) 17p
5p
: ( , )Answer p q( 5,4)
Example: Solve 1. Solve for n2. Plug into other
equation and solve for m.
3. Plug value of m into initial equation.
4. Write the ordered pair (ABC order)
2 10
2
m n
m n
2 10m n 10 2n m
2m n (10 2 ) 2m m 10 2 2m m
3 12m 4m
2 10m n 2(4) 10n
2n
: ( , )Answer m n(4,2)
Elimination1. Given two equations. Make sure variables
are lined vertically2. Choose a variable to eliminate. It must
become the opposite value of the same variable in the other equation.
3. Multiply the entire equation to create the needed values.
4. Add the two equations together5. Solve for variable left.6. Plug value into initial equation and solve.7. Write the answer as an ordered pair
Example: Solve 1. Eliminate n since they
are opposites.2. Add the two equations3. Solve for m4. Plug m back into the
initial equation5. Solve for n6. Write the ordered pair
(ABC order)
2 10
2
m n
m n
2 10
2
m n
m n
3 12m 4m
2 10m n 2(4) 10n
2n : ( , )Answer m n
(4,2)
Example: Solve 1. Eliminate h since they
are opposites. 2. Multiply the second
equation by 2.3. Add the two equations4. Solve for g5. Plug g back into the
initial equation6. Solve for h7. Write the ordered pair
(ABC order)
3 2 10
4 6
g h
g h
11 22g 2g
3 2 10g h 3(2) 2 10h
2 4h
: ( , )Answer g h
(2, 2)
3 2 10
4 6
g h
g h
3 2 10
2(4 6)
g h
g h
3 2 10
8 2 12
g h
g h
2h
Ex#1 Solve the system by substitution. Check by graphing
12 xy
2 2x y – x + 2(-2x + 1) = 2
– x – 4x + 2 = 2
-5x + 2 = 2
-2 -2
-5x = 0
-5 -5
x = 0
y = -2x + 1
y = -2(0) + 1
y = 0 + 1
y = 1
(0, 1)
(x, y)
(0, 1)
12 xy
2 2x y
2 13
4 3 11
x y
x y
Ex#2 Solve the system by substitution.
4x – 3(-2x + 13) = 11
4x + 6x – 39 = 11
10x – 39 = 11
+39 +39
10x = 50
10 10
x = 5
y = -2x + 13
y = -2(5) + 13
y = -10 + 13
y = 3
(5, 3)
(x, y)
2 13y x
Try These: Solve the system by substitution.
2 7x y 943 yx
3(2y – 7) + 4y = 9
6y – 21 + 4y = 9
10y – 21 = 9
+21 +21
10y = 30
10 10
y = 3
x = 2y – 7
x = 2(3) – 7
x = 6 – 7
x = -1
(-1, 3)
(x, y)
2 7x y
6 3 15
2 5
x y
x y
Ex#2 Solve the system by substitution.
6x – 3(2x – 5) = 15
6x – 6x + 15 = 15
15 = 15
Infinite solutions
2 5y x
3 1y x Ex#2 Solve the system by substitution.
6x + 2(-3x + 1) = 5
6x – 6x + 2 = 5
2 5
No Solution
3 1
6 2 5
x y
x y
To eliminate a variable line up the variables with x first, then y. Afterward make one of the variables opposite the other. You might have to multiply one or both equations to do this.
Example #3: Find the solution to the system using elimination.
2 6x – 2y = 16 x + 2y = 5
7x + 0 = 21
7x = 21
7 7
x = 33 + 2y = 5
-3 -3
2y = 2
y = 1 (x, y)
(3, 1)
3x – y = 8
x + 2y = 5
Example #3: Find the solution to the system using elimination.
-3 -6x – 3y = –18 4x + 3y = 24
–2x = 6
x = – 3
2(–3) + y = 6
-6 + y = 6
y = 12
(x, y)
(–3, 12)
2x + y = 6
4x + 3y = 24
3 3x – 4y = 16
5x + 6y = 14
9x – 12y = 48
2
19x = 76
x = 4
(x, y)
(4, -1)
5(4) + 6y = 14
-20 -20
6y = -6
y = -1
20 + 6y = 14
10x + 12y = 28
Example #3: Find the solution to the system using elimination.
Example #3: Find the solution to the system using elimination. 5 -3x + 2y = -10
5x + 3y = 4
-15x + 10y = -50
3
19y = -38
y = -2
(x, y)
(2, -2)
5x + 3(-2) = 4
5x = 10
x = 2
5x – 6 = 4
15x + 9y = 12
Example #3: Find the solution to the system using elimination.12x – 3y = -9
-4x + y = 3
12x – 3y = -9
3
0 = 0
Infinite solutions
-12x + 3y = 9
Example #3: Find the solution to the system using elimination.6x + 15y = -12
-2x – 5y = 9
6x + 15y = -12
3
0 = 15
No solution
-6x – 15y = 27