7.3 Solving Linear Systems of Equations
Take 3: Elimination Method
Addition/Subtraction
Can you solve the equation
2x + 3y = 12
?
Why not?
Review: Substitution Method
2x + y = 4
-x + y = 1
1. Isolate -x + y = 1
y=x+1
2. Substitute 2x + (x + 1) = 4
3. Solve 3x + 1 = 4
3x = 3
x=1
4. Plug it in y=x+1
y=1+1
y=2
(1, 2) intersecting lines
5. check -x + y = 1
2x + y = 4
-1 + 2 = 1
2(1) + 2 = 4
true!!!
true!!!
Which method(s) are correct using substitution?
A. 2x + y = 4
-x + y = 1
y = -2x + 4
-x + (-2x +4) = 1
B. 2x + y = 4
-x + y = 1
y=x+1
2x + (x + 1) = 4
C. 2x + y = 4
-x + y = 1
-x = -y + 1
-1(-x = -y + 1)
x=y-1
2(y - 1) + y = 4
Solving Systems by Elimination
1.
Rewrite system to align like terms
2.
Eliminate 1 of the variables
3.
Solve for other variable
4.
Plug it back into original equation
5.
Check
Example # 1:
-2y + x = -19
5x + 2y = 1
1. Rewrite to align like terms
x - 2y = -19
5x + 2y = 1
2. Eliminate a variable
(using addition or subtraction)
x - 2y = -19
+
5x + 2y = 1
3. Solve
6x = -18
6
6
x = -3
4. Plug it in
x - 2y = -19
-3 - 2y = -19
-2y = -16
y=8
4. Check
-2y + x = -19 -2(8) + -3 = -19
5x + 2y = 1
5(-3) + 2(8) = 1
1. Rewrite to align like terms
Example # 2:
x - 3y = -4
2. Eliminate a variable
-x - 2y = -1
3. Solve
4. Plug it in
5. Check
2x + y = 3
-x + 3y = -12
1. Rewrite to align like terms
Example # 3:
3x - 4y = -28
2. Eliminate a variable
3x + 3y = 42
3. Solve
4. Plug it in
5. Check
2x + y = 3
-x + 3y = -12
1. Rewrite to align like terms
Example # 4:
4x - y = 6
2. Eliminate a variable
2x = y
3. Solve
4. Plug it in
5. Check
2x + y = 3
-x + 3y = -12
Which solution is correct using elimination?
-3 = y + x
3x + y = 5
- 2x
( 1, -2)
-3 = y + x
3x + y = 5
x + y = -3
-(3x + y = 5)
= 2
or
x + y = -3
-(3x + y = 5)
-2x
= -8
x
= 1
1 + y = -3
x
= 4
4 + y = -3
Y = -2
y = -7
( 4, -7)