1. Warm Up
Evaluate each expression for x = 1 and y =–3.
1. x – 4y –2x + y
Write each expression in slope-intercept form.
3. y – x = 1
4. 2x + 3y = 6
5. 0 = 5y + 5x 13 –5
y = x + 1
y = x + 2
y = –x

2. Objectives
Identify solutions of linear equations in two variables.
Solve systems of linear equations in two variables by graphing.

3. Vocabulary systems of linear equations
solution of a system of linear equations

4. A system of linear equations is a set of two or more linear equations containing two or more variables. A solution of a system of linear equations with two variables is an ordered pair that satisfies each equation in the system. So, if an ordered pair is a solution, it will make both equations true.

5. Example 1A: Identifying Solutions of Systems
Tell whether the ordered pair is a solution of the given system.
(5, 2);
3x – y = 13
2 – 2 0
0 0 
3(5) –
15 – 
3x – y =13
Substitute 5 for x and 2 for y in each equation in the system.
The ordered pair (5, 2) makes both equations true.
(5, 2) is the solution of the system.

6. Example 1B: Identifying Solutions of Systems
Tell whether the ordered pair is a solution of the given system.
x + 3y = 4
(–2, 2);
–x + y = 2
–2 + 3(2) 4
x + 3y = 4 –
4 4
–x + y = 2
–(–2)
4 2
Substitute –2 for x and 2 for y in each equation in the system. 
The ordered pair (–2, 2) makes one equation true but not the other.
(–2, 2) is not a solution of the system.

7. All solutions of a linear equation are on its graph
All solutions of a linear equation are on its graph. To find a solution of a system of linear equations, you need a point that each line has in common. In other words, you need their point of intersection.
y = 2x – 1
y = –x + 5
The point (2, 3) is where the two lines intersect and is a solution of both equations, so (2, 3) is the solution of the systems.

8. Example 2A: Solving a System by Graphing
Solve the system by graphing. Check your answer.
y = x
Graph the system.
y = –2x – 3
The solution appears to be at (–1, –1).
Check
Substitute (–1, –1) into the system.
y = x
y = –2x – 3
(–1) –2(–1) –3
– – 3
–1 – 1 
y = x
(–1) (–1)
–1 –1  •
(–1, –1)
y = –2x – 3
The solution is (–1, –1).

9.   Check It Out! Example 2a
Solve the system by graphing. Check your answer.
y = –2x – 1
Graph the system.
y = x + 5
The solution appears to be (–2, 3).
Check Substitute (–2, 3) into the system.
y = x + 5
y = –2x – 1
y = –2x – 1
3 –2(–2) – 1
– 1 
y = x + 5
3 –2 + 5
3 3 
The solution is (–2, 3).

10. Example 3: Problem-Solving Application
Wren and Jenni are reading the same book. Wren is on page 14 and reads 2 pages every night. Jenni is on page 6 and reads 3 pages every night. After how many nights will they have read the same number of pages? How many pages will that be?

11. Example 3 Continued 1
Make a Plan
Write a system of equations, one equation to represent the number of pages read by each girl. Let x be the number of nights and y be the total pages read.
Total
pages is
number read
every
night
plus
already read.
Wren y = 2
 x + 14
Jenni y = 3
 x + 6

12. Example 3 Continued 2 Solve
Graph y = 2x + 14 and y = 3x + 6. The lines appear to intersect at (8, 30). So, the number of pages read will be the same at 8 nights with a total of 30 pages. 
(8, 30)
Nights

13.   Example 3 Continued Look Back 3
Check (8, 30) using both equations.
Number of days for Wren to read 30 pages.
2(8) + 14 = = 30 
Number of days for Jenni to read 30 pages.
3(8) + 6 = = 30 

14. Check It Out! Example 3
Video club A charges $10 for membership and $3 per movie rental. Video club B charges $15 for membership and $2 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost?

15. Check It Out! Example 3 Continued1
Make a Plan
Write a system of equations, one equation to represent the cost of Club A and one for Club B. Let x be the number of movies rented and y the total cost.
Total
cost is
price
for each
rental
plus
member-
ship fee.
Club A y = 3
 x + 10
Club B y = 2
 x + 15

16. Check It Out! Example 3 Continued
Solve 2
Graph y = 3x + 10 and y = 2x The lines appear to intersect at (5, 25). So, the cost will be the same for 5 rentals and the total cost will be $25.

17. Check It Out! Example 3 Continued
Look Back 3
Check (5, 25) using both equations.
Number of movie rentals for Club A to reach $25:
3(5) + 10 = = 25 
Number of movie rentals for Club B to reach $25:
2(5) + 15 = = 25 

18. Lesson Quiz: Part I
Tell whether the ordered pair is a solution of the given system.
1. (–3, 1);
2. (2, –4); no
yes

19. Lesson Quiz: Part II
Solve the system by graphing. 3.
4. Joy has 5 collectable stamps and will buy 2 more each month. Ronald has 25 collectable stamps and will sell 3 each month. After how many months will they have the same number of stamps? How many will that be?
y + 2x = 9
(2, 5)
y = 4x – 3
4 months
13 stamps

20. Make a Graphic Organizer
Must fold
Must have color
Must write out the problem and include somewhere in the organizer
Must show all 3 methods of solving the problem in the organizer
(Hint: All 3 answers should be the same!)
Be Creative!