This document discusses three methods for solving systems of equations algebraically: addition, subtraction, and substitution. Addition is used when equations contain the same term with opposite signs. Subtraction is used when equations contain the same term with the same sign. Substitution is used when addition and subtraction cannot be applied; it involves solving one equation for one variable in terms of the other and substituting into the other equation. Examples are provided for each method.

1. March 31, 2011 Samantha Billingsley
SOLVING SYSTEMS OF
EQUATIONS

2. Solving systems of equations
You can solve a system of equations with
algebra as long as there are at least the
same number of equations as variables
(for two variables, you need two
equations, etc.).
There are three ways to solve systems of
equations algebraically: addition,
subtraction, substitution.

3. Solving by addition
 We solve equations by addition when both
equations contain the same term, but with
opposite signs.
 For example: 4x + 5y = 81
- 2x - 5y = - 63
 Notice that we have the term +5y in the
first equation and -5y in the second
equation.

4. Solving by addition
To solve this system, we simply add the two
equations together like this:
4x + 5y = 81
+ (-2x -5y = -63)
2x = 18
Because +5y and -5y cancel out, we are left
with only one variable and can easily solve
the equation.
2x / 2 = 18/2
x=9

5. Solving by addition
Now that we know that x = 9, we plug this value
for x into one of our original equations:
4x + 5y = 81
4(9) + 5y = 81
36 + 5y = 81
36 - 36 + 5y = 81-36
5y = 45
5y/5 = 45/5
y=9

6. Solving by addition
 Our answer is x = 9, y=9. We can check
this by plugging both values into the
second equation:
-2x - 5y = -63
-2(9) - 5(9) = -63
-18 - 45 = -63
-63 = -63

7. Practice
Now try solving this system by addition:
2x + 3y = - 5
5x - 3y = 61

8. Solution
2x + 3y = -5
+ (5x - 3y = 61)
7x = 56
7x / 7 = 56 / 7
x=8
2(8) + 3y = -5
16 + 3y = -5
16 -16 + 3y = -5 - 16
3y = -21
3y / 3 = -21 / 3
y = -7

9. Solving by subtraction
 We can solve systems by subtraction
when both equations contain the same
term with the same sign.
 For example:
-4x - y = 11
-4x - 2y = 10
Notice that we have the term -4x in both
equations.

10. Solving by subtraction
To solve, subtract the second equation from
the first:
-4x - y = 11
- (-4x - 2y = 10)
y=1
Notice that both -4x’s cancel out when we
subtract the two equations. We are left
with just one variable, y.

11. Solving by Subtraction
Now, plug this value for y into one of the
equations.
-4x - y = 11
-4x -1 = 11
-4x -1+1 = 11+1
-4x = 12
-4x / -4 = 12 / -4
x = -3

12. Solving by Subtraction
We have x = -3, y = 1. We can check this by
plugging both values into the second
equation.
-4x - 2y = 10
-4(-3) - 2(1) =10
12 - 2 = 10
10 = 10

13. Practice
Try solving the following system by
subtraction:
x + 4y = 21
x - 3y = - 28

14. Solution
x + 4y = 21
- (x - 3y = - 28)
7y = 49
7y / 7 = 49/ 7
y=7
x + 4(7) = 21
x + 28 = 21
x + 28 - 28 = 21 - 28
x = -7

15. Solving by substitution
 If you cannot solve by addition or
subtraction, you must solve by substitution.
 Take this system for example:
x + 5y = 34
2x + 4y = 26
First, solve one equation for one variable
(leaving it in terms of the other variable). In
this case, we will solve the first equation for
x, in terms of y.
x + 5y - 5y = 34 - 5y
x = 34 - 5y

16. Solving by substitution
Next, substitute your solution into the second equation.
x = 34 - 5y
2x + 4y = 26
2(34 - 5y) + 4y = 26
Using the distributive property, we get:
68 -10y + 4y = 26
68 -6y = 26
68 - 68 -6y = 26 – 68
-6y = -42
-6y / -6 = -42 / -6
y=7

17. Solving by substitution
Now that we know y = 7, we can plug this
value into our previous solution for x.
x = 34 - 5y
x = 34 - 5(7)
x = 34 - 35
x = -1

18. Solving by substitution
 We have x = -1, y = 7. To check, plug both
values into one of the original equations.
2x + 4y = 26
2(-1) + 4(7) = 26
-2 + 28 = 26
26 = 26

19. Practice
Try solving the following system by
substitution:
y=-x+3
- 5x = - 43 + y

20. Solution
y=-x+3
- 5x = - 43 + y
-5x = - 43 + (-x + 3)
-5x = -43 - x + 3
-5x = -40 - x
-5x + x = -40 -x + x
-4x = -40
-4x / -4 = -40 / -4
x =10
y = -10 + 3
y = -7

21. Review
- 5x + 3y = 1
- 4x + 3y = 5
Addition, subtraction or substitution? Why?
Subtraction, because both equations have
the term +3y.

22. Review
4x = - 2y + 56
x = - 5y + 59
Addition, subtraction or substitution? Why?
Substitution, because the equations have no
terms with the same number.

23. Review
- x + y = 12
x - 3y = - 30
Addition, subtraction or substitution? Why?
Addition, because the first equation has -x
and the second has x.

24. THE END