1. Learn to solve systems of equations.

2. A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.

3. Example 1: Identifying Solutions of a System of Equations
Determine if the ordered pair is a solution of the system of equations below.
5x + y = 7
x – 3y = 11
1. (1, 2)
5x + y = 7
x – 3y = 11
Substitute for x and y.
5(1) + 2 = 7 ?
1 – 3(2) = 11 ?
7 = 7 
–5  11 
The ordered pair (1, 2) is not a solution of the system of equations.

4. Example 2: Identifying Solutions of a System of Equations
Determine if the ordered pair is a solution of the system of equations below.
5x + y = 7
x – 3y = 11
2. (2, –3)
5x + y = 7
x – 3y = 11
5(2) + –3 = 7 ?
Substitute for x and y.
2 – 3(–3) = 11 ?
7 = 7
11 = 11  
The ordered pair (2, –3) is a solution of the system of equations.

5. Example 3
Determine if each ordered pair is a solution of the system of equations below.
4x + y = 8
x – 4y = 12
3. (1, 2)
4x + y = 8
x – 4y = 12
Substitute for x and y.
4(1) + 2 = 8 ?
1 – 4(2) = 12 ?
6  8 
–7  12 
The ordered pair (1, 2) is not a solution of the system of equations.

6. Example 4
Determine if each ordered pair is a solution of the system of equations below.
4x + y = 8
x – 4y = 12
4. (1, 4)
4x + y = 8
x – 4y = 12
Substitute for x and y.
4(1) + 4 = 8 ?
1 – 4(4) = 12 ?
8 = 8 
–15  12 
The ordered pair (1, 4) is not a solution of the system of equations.

7. When solving systems of equations, remember to find values for all of the variables.
Helpful Hint

8. Example 5: Solving Systems of Equations
y = x – 4
Solve the system of equations.
y = 2x – 9
y = y
y = x – 4
y = 2x – 9
x – 4 = 2x – 9
Solve the equation to find x.
x – 4 = 2x – 9
– x – x
Subtract x from both sides.
–4 = x – 9
Add 9 to both sides.
5 = x

9. Example 5 Continued
To find y, substitute 5 for x in one of the original equations.
y = x – 4 = 5 – 4 = 1
The solution is (5, 1).
Check: Substitute 5 for x and 1 for y in each equation.
y = x – 4
y = 2x – 9
1 = 2(5) – 9 ?
1 = 5 – 4 ?
1 = 1 
1 = 1 

10. Example 6
y = x – 5
Solve the system of equations.
y = 2x – 8
y = y
y = x – 5
y = 2x – 8
x – 5 = 2x – 8
Solve the equation to find x.
x – 5 = 2x – 8
– x – x
Subtract x from both sides.
–5 = x – 8
Add 8 to both sides.
3 = x

11. Example 6 Continued
To find y, substitute 3 for x in one of the original equations.
y = x – 5 = 3 – 5 = –2
The solution is (3, –2).
Check: Substitute 3 for x and –2 for y in each equation.
y = x – 5
y = 2x – 8
–2 = 2(3) – 8 ?
–2 = 3 – 5 ?
–2 = –2 
–2 = –2 

12. To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

13. You can choose either variable to solve for
You can choose either variable to solve for. It is usually easiest to solve for a variable that has a coefficient of 1.
Helpful Hint

14. Example 7: Solving Systems of Equations
Solve the system of equations.
7. 3x – 3y = x + y = -5
Solve both equations for y.
3x – 3y = – x + y = –5
–3x –3x –2x –2x
–3y = –3 – 3x y = –5 – 2x –3 3x
–3y
= –
y = 1 + x
1 + x = –5 – 2x

15. Example 10 Continued
1 + x = –5 – 2x
Add 2x to both sides.
+ 2x x
1 + 3x = –5
Subtract 1 from both sides.
– –1
3x = –6
Divide both sides by 3. –6 3 3x =
x = –2
y = 1 + x
Substitute –2 for x.
= 1 + –2 = –1
The solution is (–2, –1).

16. Example 11
Solve the system of equations.
11. x + y = x + y = –1
Solve both equations for y.
x + y = x + y = –1
–x –x – 3x – 3x
y = 5 – x y = –1 – 3x
5 – x = –1 – 3x
Add x to both sides.
+ x x
= –1 – 2x

17. Example 11 Continued
= –1 – 2x
Add 1 to both sides.
= –2x
–3 = x
Divide both sides by –2.
y = 5 – x
= 5 – (–3) Substitute –3 for x.
= = 8
The solution is (–3, 8).

18. Example 12
Solve the system of equations.
12. x + y = – –3x + y = 2
Solve both equations for y.
x + y = – –3x + y = 2
– x – x x x
y = –2 – x y = 2 + 3x
–2 – x = 2 + 3x

19. Example 12 Continued
–2 – x = 2 + 3x
Add x to both sides.
+ x x
– = 2 + 4x
Subtract 2 from both sides.
– –2
– = x
Divide both sides by 4.
–1 = x
y = 2 + 3x
Substitute –1 for x.
= 2 + 3(–1) = –1
The solution is (–1, –1).

20. Lesson Review
1. Determine if the ordered pair (2, 4) is a solution of the system. y = 2x; y = –4x + 12
Solve each system of equations.
2. y = 2x + 1; y = 4x
3. 6x – y = –15; 2x + 3y = 5
4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers.
yes
( , 2) 1 2
(–2,3)
15 and 8