Download Simplify the following expressions: 1. 3(3 − 5 ) 2. -( + 3 ) 3.

Warm Up 11-20
Simplify the following expressions:
1. 3(3𝑥 − 5𝑦)
2. -(𝑦 + 3𝑥)
3. -6(2𝑥 + 6𝑦)
Solve the System of Equations:
1
4. y = 2𝑥 − 3
5. y = − 𝑥 − 2
3
2y − 3𝑥 = −4
y=𝑥+2
Unit 5 Day 3 Solving Systems
by Linear Combinations
Essential Questions: How can we solve systems using the
linear combinations method?
Vocabulary
• Opposite Coefficients: two terms that have the same
numerical coefficient but with opposite signs
• Example: 2x and -2x
Solving Systems Using
Linear Combinations
1. Make sure your equations are lined up.
2. If necessary, multiply one or both equations by a
number to get opposite coefficients.
3. Combine the equations together (one variable
should be eliminated) and solve for the remaining
variable.
4. Plug your answer into one of your original
equations to solve for the other variable.
Example 1: Use linear combinations to solve the linear system.
-x + 2y = -8
x
+
6y
=
-16
+
8y = -24
x + 6y = -16
x + 6(-3) = -16
x -18 = -16
+ 18 + 18
x=2
Solution: (2, -3)
8
8
y = -3
Check!
-x + 2y = -8
-(2) + 2(-3) = -8
-2 - 6 = -8
-8 = -8
x + 6y = -16
2 + 6(-3) = -16
2 - 18 = -16
-16 = -16
Example 2: Use linear combinations to solve the linear system.
3x + 2y = 8
+ -5x - 2y = 12
-2x = 20
3x + 2y = 8
3(-10) + 2y = 8
-30 + 2y = 8
+ 30
+ 30
-2
-2
x = -10
Solution: (-10, 19)
2y = 38
2
2
y = 19
Don't forget to check your solution
in both original equations!!
Example 3: Use linear combinations to solve the linear system.
2(9x - 3y = 20 )
3x + 6y = 2
3x + 6y = 2
3(2) + 6y = 2
6 + 6y = 2
-6
-6
18x - 6y = 40
+ 3x + 6y = 2
21x = 42
21
x=2
6y = -4
6
6
2
y=- 3
21
2
Solution: (2, )
3
Don't forget to
check your
solution in
both original
equations!!
Example 4: Use linear combinations to solve the linear system.
3x + 2y = 46
-3( x + 5y = 11 )
3x + 2y = 46
3x + 2(-1) = 46
3x - 2 = 46
+
2
+
2
+
3x = 48
3
3
x = 16
3x + 2y = 46
+ -3x - 15y = -33
-13y = 13
-13 -13
y = -1
Solution: (16, -1)
Don't forget to
check your
solution in
both original
equations!!
Example 5: Use linear combinations to solve the linear system.
-3( 2x - 3y = 0 )
2( 3x - 2y = 5 )
-6x + 9y = 0
+ 6x - 4y = 10
5y = 10
2x - 3y = 0
2x - 3(2) = 0
2x - 6 = 0
+6 +6
2x = 6
x=3
5
5
y=2
Solution: (3, 2)
Don't forget to
check your
solution in
both original
equations!!
Example 6: Use linear combinations to solve the linear
system.
-1( 2x + y = 5 )
-2x - y = -5
2x + y = 1
+ 2x + y = 1
0 = -4
No Solution!
Example 7: Use linear combinations to solve the linear
system.
-2( -2x + y = 3 )
4x- 2y = -6
-4x + 2y = 6
+ -4x + 2y = 6
0=0
All Real Numbers!
Summary
Essential Questions: How can we solve systems using
the linear combinations method?
Take one minute to write two sentences answering
the essential questions.