1. Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes

2. Solve for the indicated variable. 1. P = R – C for R 2. V = Ah for A
Warm Up
Solve for the indicated variable.
1. P = R – C for R
2. V = Ah for A
3. R = for C
R = P + C 1 3
= A 3V h
C – S t
Rt + S = C

3. Problem of the Day
At an audio store, stereos have 2 speakers and home-theater systems have 5 speakers. There are 30 sound systems with a total of 99 speakers. How many systems are stereo systems and how many are home-theater systems?
17 stereo systems, 13 home-theater systems

4. Learn to solve systems of equations.

5. Vocabulary
system of equations
solution of a system of equations

6. 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.

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

8. Additional Example 1A: Solving Systems of Equations
Solve the system of equations.
y = 4x – 6
y = x + 3
The expressions x + 3 and 4x – 6 both equal y. So by the Transitive Property they are equal to each other.
y = 4x – 6
y = x + 3
4x – 6 = x + 3

9. Additional Example 1A Continued
Solve the equation to find x.
4x – 6 = x + 3
– x – x
Subtract x from both sides.
3x – 6 =
Add 6 to both sides. 3x
Divide both sides by 3. =
x =
To find y, substitute 3 for x in one of the original equations.
y = x + 3 = = 6
The solution is (3, 6).

10. Additional Example 1B: Solving Systems of Equations
y = 2x + 9
y = –8 + 2x
2x + 9 = –8 + 2x
Transitive Property
Subtract 2x from both sides.
– 2x – 2x
9 ≠ –8
The system of equations has no solution.

11. Check It Out: Example 1A
Solve the system of equations.
y = x – 5
y = 2x – 8
The expressions x – 5 and 2x – 8 both equal y. So by the Transitive Property they equal each other.
y = x – 5
y = 2x – 8
x – 5 = 2x – 8

12. Check It Out: Example 1A Continued
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
To find y, substitute 3 for x in one of the original equations.
y = x – 5 = 3 – 5 = –2
The solution is (3, –2).

13. Check It Out: Example 1B
y = 3x – 7
y = 6 + 3x
3x – 7 = 6 + 3x
Transitive Property
Subtract 3x from both sides.
– 3x – 3x
–7 ≠ 6
The system of equations has no solution.

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

15. Additional Example 2A: Solving Systems of Equations by Solving for a Variable
Solve the system of equations.
5x + y = x – 3y = 11
Solve both equations for x.
5x + y = x – 3y = 11
– y – y y y
5x = 7 – y x = y
5(11 + 3y)= 7 – y
y = 7 – y
Subtract 15y from both sides.
– 15y – 15y
= 7 – 16y

16. Additional Example 2A Continued
= 7 – 16y
Subtract 7 from both sides.
– –7
Divide both sides by –16.
– 16y
– = – 16
– = y
x = y
= (–3) Substitute –3 for y.
= 11 + –9 = 2
The solution is (2, –3).

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

18. Additional Example 2B: Solving Systems of Equations by Solving for a Variable
Solve the system of equations.
–2x + 10y = – x – 5y = 4
Solve both equations for x.
–2x + 10y = – x – 5y = 4
–10y –10y y y
–2x = –8 – 10y x = 4 + 5y
= – –8 –2
10y
–2x
x = 4 + 5y
Subtract 5y from both sides.
4 + 5y = 4 + 5y
– 5y – 5y
4 = 4
Since 4 = 4 is always true, the system of equations has an infinite number of solutions.

19. Check It Out: Example 2A
Solve the system of equations.
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

20. Check It Out: Example 2A Continued
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).

21. Check It Out: Example 2B
Solve the system of equations.
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

22. Check It Out: Example 2B 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).

23. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

24. ( , 2) Lesson Quiz Solve each system of equations. 1. y = 5x + 10
3. 6x – y = –15
2x + 3y = 5
4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers.
no solution
( , 2) 1 2
(–2, 3)
15 and 8

25. Lesson Quiz for Student Response Systems
1. Solve the given system of equations.
y = 11x + 20
y = –2 + 11x
A. (2, 2)
B. (1, 1)
C. (1, –1)
D. no solution

26. Lesson Quiz for Student Response Systems
2. Solve the given system of equations.
4x + y = 11
2x + 3y = –7
A. (4, –5)
B. (4, 5)
C. (2, –5)
D. (2, 5) 26

27. Lesson Quiz for Student Response Systems
3. Two numbers have a sum of 37 and a difference of 17. Identify the two numbers.
A. –27 and –10
B. –27 and 10
C. 27 and 10
D. 27 and –10 27