1. Systems of Equations 11-6 Warm Up Problem of the Day
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. Systems of Equations 11-6 Warm Up Solve for the indicated variable.
Course 3
11-6
Systems of Equations
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. Systems of Equations 11-6 Problem of the Day
Course 3
11-6
Systems of Equations
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. Systems of Equations Learn to solve systems of equations. 11-6
Course 3
11-6
Systems of Equations
Learn to solve systems of equations.

5. Insert Lesson Title Here
Course 3
11-6
Systems of Equations
Insert Lesson Title Here
Vocabulary
system of equations
solution of a system of equations

6. Course 3
11-6
Systems of Equations
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. Systems of Equations 11-6 Caution!
Course 3
11-6
Systems of Equations
When solving systems of equations, remember to find values for all of the variables.
Caution!

8. Additional Example 1A: Solving Systems of Equations
Course 3
11-6
Systems of Equations
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 equal each other.
y = 4x – 6
y = x + 3
4x – 6 = x + 3

9. Additional Example 1A Continued
Course 3
11-6
Systems of Equations
Additional Example 1A Continued
Solve the equation to find x.
4x – 6 = x + 3
– x – x
Subtract x from both sides.
3x =
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
Course 3
11-6
Systems of Equations
Additional Example 1B: Solving Systems of Equations
y = 2x + 9
y = x
2x + 9 = x
Transitive Property
Subtract 2x from both sides.
– 2x – 2x
9 ≠ -8
The system of equations has no solution.

11. Systems of Equations 11-6 Check It Out: Example 1A
Course 3
11-6
Systems of Equations
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
Course 3
11-6
Systems of Equations
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. Systems of Equations 11-6 Check It Out: Example 1B y = 3x + -7
Course 3
11-6
Systems of Equations
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. Course 3
11-6
Systems of Equations
To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

15. Course 3
11-6
Systems of Equations
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 y
= 7 – 16y

16. Additional Example 2A Continued
Course 3
11-6
Systems of Equations
Additional Example 2A Continued
= 7 – 16y
Subtract 7 from both sides.
– –7 y
– =
Divide both sides by –16.
= y
x = y
= (-3) Substitute –3 for y.
= 11 + –9 = 2
The solution is (2, –3).

17. Systems of Equations 11-6 Helpful Hint
Course 3
11-6
Systems of Equations
You can solve for either variable. It is usually easiest to solve for a variable that has a coefficient of 1.
Helpful Hint

18. Course 3
11-6
Systems of Equations
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 y
4 = 4
Since 4 = 4 is always true, the system of equations has an infinite number of solutions.

19. Systems of Equations 11-6 Check It Out: Example 2A
Course 3
11-6
Systems of Equations
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
Course 3
11-6
Systems of Equations
Check It Out: Example 2A 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).

21. Systems of Equations 11-6 Check It Out: Example 2B
Course 3
11-6
Systems of Equations
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
Course 3
11-6
Systems of Equations
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. Insert Lesson Title Here
Course 3
11-6
Systems of Equations
Insert Lesson Title Here
Lesson Quiz
Solve each system of equations.
1. y = 5x + 10
y = –7 + 5x
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.
no solution
( , 2) 1 2
(–2,3)
15 and 8