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Objective: To identify and write linear systems of equations.

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**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?

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**“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.

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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.

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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)?

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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

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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.

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**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

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**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.

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**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.

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**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

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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.

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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)

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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)

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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)

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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.

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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

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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.

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**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.

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