1. Objective:
To identify and write linear systems of equations.

2. A shipping clerk must send packages of chemicals to a laboratory
A shipping clerk must send packages of chemicals to a laboratory. The container she is using to ship the chemicals will hold 18 lbs. Each package of chemicals weighs 3 lbs. How many packages of chemicals will the container hold?

3. “Systems of equations” just means that we are dealing with more than one EQUATION and VARIABLE. 
So far, we’ve basically just played around with the equation for a line, which is  y=mx + b.

4. Black Friday Shopping
You want to buy some chocolate candy for your math teacher (ehem). The first website you find (chocolateisamazing.com) charges $3 plus $1 per pound to ship a box. The second website (hersheycandyisthebest.com) charges $1 plus $2 per pound to ship the same item. What equations can you write to represent the situation? Answer: For an object that weighs x pounds, the charges for the two websites are represented by the equations y = x + 3 and y = 2x + 1.

5. You are going to the mall with your friends and you have $200 to spend from your recent birthday (or Christmas) money. You discover a store that has all jeans for $25 and all dresses for $50.  You really, really want to take home 6 items of clothing because you “need” that many new things.
Wouldn’t it be clever to find out how many pairs of jeans and how many dresses you can buy so you use the whole $200 (tax not included – your parents promised to pay the tax)?

6. So what we want to know is how many pairs of jeans we want to buy (let’s say “j”) and how many dresses we want to buy (let’s say “d”). So always write down what your variables will be:
Let j = the number of
jeans you will buy
Let d = the number
of dresses you’ll buy

8. System of Equations:
Two or more equations with the same set of variables are called a system of equations.
A solution of a system of equations is an ordered pair that satisfies each equation in the system.

9. Check the intersection point
EXAMPLE 1
Check the intersection point
Use the graph to solve the system. Then check your solution algebraically.
x + 2y = 7
Equation 1
3x – 2y = 5
Equation 2
SOLUTION
The lines appear to intersect at the point (3, 2).
CHECK
Substitute 3 for x and 2 for y in each equation.
x + 2y = 7
3 + 2(2) = ? 7
7 = 7

10. Check the intersection point
EXAMPLE 1
Check the intersection point
3x – 2y = 5
3(3) – 2(2) 5 = ?
5 = 5
ANSWER
Because the ordered pair (3, 2) is a solution of each equation, it is a solution of the system.

11. Use the graph-and-check method
EXAMPLE 2
Use the graph-and-check method
Solve the linear system:
–x + y = –7
Equation 1
x + 4y = –8
Equation 2
SOLUTION
STEP 1
Graph both equations.

12. Use the graph-and-check method
EXAMPLE 2
Use the graph-and-check method
STEP 2
Estimate the point of intersection. The two lines appear to intersect at (4, – 3).
STEP 3
Check whether (4, –3) is a solution by substituting 4 for x and –3 for y in each of the original equations.
Equation 1
Equation 2
–x + y = –7
x + 4y = –8
–(4) + (–3) –7 = ?
4 + 4(–3) –8 = ?
–7 = –7
–8 = –8

13. EXAMPLE 2
Use the graph-and-check method
ANSWER
Because (4, –3) is a solution of each equation, it is a solution of the linear system.

14. EXAMPLE 2
GUIDED PRACTICE
Use the graph-and-check method
for Examples 1 and 2
Solve the linear system by graphing. Check your solution.
–5x + y = 0 1.
5x + y = 10
ANSWER
(1, 5)

15. EXAMPLE 2
GUIDED PRACTICE
Use the graph-and-check method
for Examples 1 and 2
Solve the linear system by graphing. Check your solution.
2x + y = 4
–x + 2y = 3 2.
ANSWER
(1, 2)

16. EXAMPLE 2
GUIDED PRACTICE
Use the graph-and-check method
for Examples 1 and 2
Solve the linear system by graphing. Check your solution.
3x + y = 3
x – y = 5 3.
ANSWER
(2, 3)

17. EXAMPLE 3
Standardized Test Practice
The parks and recreation department in your town offers a season pass for $90. •
As a season pass holder, you pay $4 per session to use the town’s tennis courts. •
Without the season pass, you pay $13 per session to use the tennis courts.

18. GUIDED PRACTICE
for Example 3
4. Solve the linear system in Example 3 to find the number of sessions after which the total cost with a season pass, including the cost of the pass, is the same as the total cost without a season pass.
ANSWER
10 sessions

19. EXAMPLE 4
Solve a multi-step problem
RENTAL BUSINESS
A business rents in-line skates and bicycles. During one day, the business has a total of 25 rentals and collects $450 for the rentals. Find the number of pairs of skates rented and the number of bicycles rented.

20. Solve a multi-step problem
EXAMPLE 4
Solve a multi-step problem
STEP 3
Estimate the point of intersection. The two lines appear to intersect at (20, 5).
STEP 4
Check whether (20, 5) is a solution. = ?
15(20) + 30(5) = ?
25 = 25
450 = 450
ANSWER
The business rented 20 pairs of skates and 5 bicycles.