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. Sunshine State Standards
MA.8.A.1.3 Use tables, graphs, and models to represent, analyze, and solve real-world problems related to systems of linear equations.

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

6. 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

7. 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).

8. 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.

9. Check It Out: Example 1A
Solve each system of equations.
y = x – 5
y = 2x – 8
x – 5 = 2x – 8
y = x – 5
3 = x
y = (3) – 5 = –2
The solution is (3, –2).

10. Check It Out: Example 1B
Solve each system of equations.
y = 2x
y = x + 6
2x = x + 6
y = 2x
x = 6
y = 2(6) = 12
The solution is (6, 12).

11. When equations in a system are not already solve for one variable, you can solve both equations for x or both for y.

12. Additional Example 2A: Solving Systems of Equations by Solving for a Variable
Solve the system of equations.
x + 4y = – x – 3y = 11
Solve both equations for x.
x + 4y = – x – 3y = 11
–4y –4y y y
x = –10 – 4y x = y
–10 – 4y = y
–10 – 4y = y
Subtract 3y from both sides.
– 3y – 3y
–10 – 7y = 11

13. Additional Example 2A Continued
–10 – 7y = 11
Add 10 to both sides.
– 7y
Divide both sides by –7.
–7 = – 7
y = –3
x = y
= (–3) Substitute –3 for y.
= 11 + –9 = 2
The solution is (2, –3).

14. 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

15. 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.

16. Check It Out: Example 2A
Solve each system of equations.
2x + y = 0
2x + 3y = 8
−2x + 8 3
y = −2x and y =
−2x + 8 3
–2x = ; x –2
2x + y = 0
2(−2) + y = 0
y = 4
The solution is (–2, 4).

17. Check It Out: Example 2B
Solve the system of equations.
y = x –1
–3x + 3y = 4
3x + 4 3
y = x − 1 and y =
3x + 4 3
x – 1 =
−3 ≠ 4
There are no solutions.

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

19. ( , 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

20. 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

21. 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) 21

22. 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 22