7.2, 7.3 Solving by Substitution
and Elimination
Solving by Substitution
1) Use one equation to get either x or y by itself
2) Substitute what we just found in step 1 to
the other equation
3) Solve for one unknown
4) Substitute the answer from step 3 to either
of the original equation to solve for the
other unknown
5) Check your answer
x+y=4
x=y+2
Step 1) x is already by itself x = y + 2
Step 2) Substitute to the first equation
x +y=4
(y + 2) + y = 4
Step 3)
2y + 2 = 4
2y
=4–2=2
y
=1
Step 4) x = y + 2
x=1+2=3
Solution: (3, 1)
2x + y = 6
3x + 4y = 4
Step 1) get y by itself 2x + y = 6 so y = -2x + 6
Step 2) Substitute to the second equation
3x + 4y = 4
3x + 4(-2x + 6) = 4
Step 3) 3x – 8x + 24 = 4
-5x + 24
=4
-5x
= -20
x
=4
Step 4) 2x + y = 6
2(4)+y = 6 so 8 + y = 6, y = -2
Solution: (4, -2)
More practice: (Make sure you check your answers)
1) 3x – 4y = 14
5x + y = 8
Answer: (2,-2)
2) 2x – 3y = 0
-4x + 3y = -1
Get x by itself: 2x – 3y = 0 so x = (3/2) y
Substitute into: -4x
+ 3y = -1
-4(3/2)y + 3y = -1
-6y + 3y = -1
- 3y = -1 so y = 1/3
Solve for x: Use the first equation 2x – 3y
=0
2x – 3(1/3) = 0
2x
–1
=0
x
=½
Answer: (1/2 , 1/3)
Example 1
• The perimeter of a basketball court is 288
ft. The length is 40 ft longer than the
width. Find the dimensions of the court.
Example 2
• The sum of two numbers is 51. One
number is 27 more than the other. Find the
numbers.
Solve by Elimination
1) Multiply some numbers to either or both
equations to get 2 opposite terms (For
example: 2x and -2x)
2) Add equations to eliminate one variable
3) Solve for 1 unknown
4) Substitute the answer from step 3 to either of
the original equation to solve for the other
unknown
5) Check your answer
• Solve by elimination
2x – 3y = 0
-4x + 3y = -1
Ignore step 1 since we already have 2 opposite terms -3y and 3y
Step 2 &3: Add 2 equations to eliminate y and solve for x
2x – 3y = 0
-4x + 3y = -1
-2x
= -1
x
= 1/2
Step 4: Choose the first equation 2x – 3y = 0
2(1/2) – 3y = 0
1
- 3y = 0
- 3y = -1
y = 1/3
Answer: ( 1/2, 1/3)
Ex2) 2x + 2y = 2
3x – y = 1
Step 1: Multiply 2 to the second equation
2x + 2y = 2
(2) 3x – y = 1
2x + 2y = 2
6x – 2y = 2
8x
= 4 (step 2 and 3)
x
= 1/2
Step 4: Choose 3x – y = 1
3(1/2) – y = 1
3/2 - y = 1
- y = 1 – (3/2) = - ½
y=½
Answer: (1/2, 1/2)
Ex3) 2x + 3y = 8
-3x + 2y = 1
Step 1: Multiply 3 to the 1st equation and 2 to the 2nd equation
(3) 2x + 3y = 8
(2) -3x + 2y = 1
6x + 9y = 24
-6x + 4y = 2
13y = 26 (step 2 and 3)
y =2
Step 4: Choose -3x + 2y = 1
-3x + 2(2) = 1
-3x + 4 = 1
-3x = 1 -4 = -3
x=1
Answer: (1,2)
Practice
Solve by elimination
1) 3x – 4y = 14
5x + y = 8
Answer: (2,-2)
•
3x + 2y = 7
6x + 4y = 14
Answer: Infinite many solutions