Presentation on theme: "Objective: To identify and write linear systems of equations."— Presentation transcript:
1Objective:To identify and write linear systems of equations.
2A 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.
4Black Friday ShoppingYou 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.
5You 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)?
6So 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 ofjeans you will buyLet d = the numberof dresses you’ll buy
8System 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.
9Check the intersection point EXAMPLE 1Check the intersection pointUse the graph to solve the system. Then check your solution algebraically.x + 2y = 7Equation 13x – 2y = 5Equation 2SOLUTIONThe lines appear to intersect at the point (3, 2).CHECKSubstitute 3 for x and 2 for y in each equation.x + 2y = 73 + 2(2)=?77 = 7
10Check the intersection point EXAMPLE 1Check the intersection point3x – 2y = 53(3) – 2(2)5=?5 = 5ANSWERBecause the ordered pair (3, 2) is a solution of each equation, it is a solution of the system.
11Use the graph-and-check method EXAMPLE 2Use the graph-and-check methodSolve the linear system:–x + y = –7Equation 1x + 4y = –8Equation 2SOLUTIONSTEP 1Graph both equations.
12Use the graph-and-check method EXAMPLE 2Use the graph-and-check methodSTEP 2Estimate the point of intersection. The two lines appear to intersect at (4, – 3).STEP 3Check whether (4, –3) is a solution by substituting 4 for x and –3 for y in each of the original equations.Equation 1Equation 2–x + y = –7x + 4y = –8–(4) + (–3) –7=?4 + 4(–3) –8=?–7 = –7–8 = –8
13EXAMPLE 2Use the graph-and-check methodANSWERBecause (4, –3) is a solution of each equation, it is a solution of the linear system.
14EXAMPLE 2GUIDED PRACTICEUse the graph-and-check methodfor Examples 1 and 2Solve the linear system by graphing. Check your solution.–5x + y = 01.5x + y = 10ANSWER(1, 5)
15EXAMPLE 2GUIDED PRACTICEUse the graph-and-check methodfor Examples 1 and 2Solve the linear system by graphing. Check your solution.2x + y = 4–x + 2y = 32.ANSWER(1, 2)
16EXAMPLE 2GUIDED PRACTICEUse the graph-and-check methodfor Examples 1 and 2Solve the linear system by graphing. Check your solution.3x + y = 3x – y = 53.ANSWER(2, 3)
17EXAMPLE 3Standardized Test PracticeThe 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.
18GUIDED PRACTICEfor Example 34. 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.ANSWER10 sessions
19EXAMPLE 4Solve a multi-step problemRENTAL BUSINESSA 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.
20Solve a multi-step problem EXAMPLE 4Solve a multi-step problemSTEP 3Estimate the point of intersection. The two lines appear to intersect at (20, 5).STEP 4Check whether (20, 5) is a solution.=?15(20) + 30(5)=?25 = 25450 = 450ANSWERThe business rented 20 pairs of skates and 5 bicycles.