1. SYSTEMS OF EQUATIONS

2. Substitution Method Ex. 1: 2x + y = 15 y = 3x so 5x = 15 and x = 3
You now have to go back to
one of your original equations
where you will substitute the
known value of x in
order to determine the value of y
We have been using the
Substitution Property in our proofs.
Now we are going to
use substitution to solve
systems of equations.
2x + 3x = 15
so 5x = 15
and x = 3
y = 3(3) = 9
Write your answer as an ordered pair (3, 9)

3. Ex. 2 3x – y = 13 y = x + 5 3x – ( x + 5 ) = 13 3x – x – 5 = 13
Now substitute back into one of the original equations to find y. y = = 14

4. You may have to rearrange one of the equations before you do the substitution.
x + 4y = 20
2x + 3y = 10
Rearrange:
x = 20 – 4y
Now substitute and solve for y:
2 (20 – 4y)+3y=10 so 40 – 8y + 3y=10
-5y = so y = 6
Find x: x + 4(6) = 20
x+ 24 = 20
x = -4

5. If you add these two equations together, the y term drops out!
Solving systems of equations by adding or subtracting equations
Example: x + 5y = -7
3x – 5y = 15
Now substitute into either equation to find y:
2 + 5y = -7 5y = -9
y = -9/5
If you add these two equations together, the y term drops out!
4x = 8
x = 2

6. Example: 4x – y = 16
3x – y = 11
x = 5
4(5) – y = 16
20 – y = 16
y = 4
(5, 4)

7. After substitution we find y = 14 (6,14)
You may have to multiply one of the equations before you add or subtract:
-3x + 2y = 10
2x – y = -2
Multiply this
equation
by 2
4x – 2y = -4
Now add
the
equations
together
x = 6
After substitution we find y = 14 (6,14)

8. You may have to multiply both of the equations by a number to get a variable to cancel out.
2x + 5y = 36
3x – 2y = -3
Multiply by 3
6x + 15y = 108
Multiply by 2
6x - 4y = -6
19y = 114
y= 6