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Do Now 1/10/11 Copy HW in your planner. Copy HW in your planner. Text p. 430, #4-20 evens, 30-34 evens Text p. 430, #4-20 evens, 30-34 evens Text p. 439, #4-24 even, #32, #36 Text p. 439, #4-24 even, #32, #36 Open your textbook to page 424 and preview Chapter 7 “Systems of Equations and Inequalities” Open your textbook to page 424 and preview Chapter 7 “Systems of Equations and Inequalities”

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Chapter 7 Preview “Solving and Graphing Linear Systems” (7.1) Solve Linear Systems by Graphing (7.2) Solve Linear Systems by Substituting (7.3) Solve Linear Systems by Adding or Subtracting (7.4) Solve Linear Systems by Multiplying First (7.5) Solve Special Types of Linear Systems (7.6) Solve Systems of Linear Inequalities

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Linear System– consists of two or more linear equations. consists of two or more linear equations. Equation 1 Equation 1 3x – 2y = 5 3x – 2y = 5 Equation 2 Equation 2 x + 2y = 7 x + 2y = 7 Section 7.1 “Solve Linear Systems by Graphing” A solution to a linear system is an ordered pair (a point) where the two linear equations (lines) intersect (cross).

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Solving a Linear System by Graphing (1) Graph both equations in the same plane. (2) Estimate the coordinates of the point where the two lines intersect. (3) Check the coordinate by substituting into EACH equation of the linear system, to see if the point is a solution for both equations.

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7 = 7 SOLUTION Use the graph to solve the system. Then check your solution algebraically. x + 2y = 7 Equation 1 3x – 2y = 5 Equation 2 The lines appear to intersect at the point (3, 2). CHECK Substitute 3 for x and 2 for y in each equation. x + 2y = 7x + 2y = 7x + 2y = 7x + 2y = 7 3 + 2(2) =?7 3x – 2y = 53x – 2y = 53x – 2y = 53x – 2y = 5 5 = 5 3(3) – 2(2 ) 5 =? Equation 1 Equation 2 Because the ordered pair (3, 2) is a solution of each equation, it is a solution of the system. Using a Graph to Solve a Linear System

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SOLUTION Use the graph to solve the system. Then check your solution algebraically. x + 4y = -8 Equation 1 -x + y = -7 Equation 2 The lines appear to intersect at the point (4, -3). CHECK Substitute 4 for x and -3 for y in each equation. x + 4y = -8 4 + 4(-3) =?-8 -x + y = -7 -7= -7 -4 + (-3 ) -4 + (-3 ) -7 =? Equation 1 Equation 2 Because the ordered pair (4, -3) is a solution of each equation, it is a solution of the system. Using a Graph to Solve a Linear System -8 = -8

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Standardized Test Practice Which system of equations can be used to find the number x of sessions of tennis after which the total cost y with a season pass, including the cost of the pass, is the same as the total cost without a season pass? y = 13x y = 13x y = 4x y = 4x A y = 13x y = 13x y = 90 + 4x y = 90 + 4x C y = 4x y = 90 + 13x B y = 90 + 4x y = 90 + 13x D 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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Standardized Test Practice Which system of equations can be used to find the number x of sessions of tennis after which the total cost y with a season pass, including the cost of the pass, is the same as the total cost without a season pass? y = 13x y = 13x y = 4x y = 4x A y = 13x y = 13x y = 90 + 4x y = 90 + 4x C y = 4x y = 90 + 13x B y = 90 + 4x y = 90 + 13x D EQUATION 1 y = 13 x EQUATION 2 y = 90 + 4 x

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Solve a multi-step problem 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. STEP 1 Write a linear system. Let x be the number of pairs of skates rented, and let y be the number of bicycles rented. x + y =25 15x + 30y = 450 Equation for number of rentals Equation for money collected from rentals

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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. 20 + 5 25 =? 15( 20 ) + 30(5) 450 =? 450 = 450 25 = 25 ANSWER The business rented 20 pairs of skates and 5 bicycles. STEP 2 Graph both equations.

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Solving a Linear System by Substitution (1) Solve one of the equations for one of its variables. (When possible, solve for a variable that has a coefficient of 1 or -1). (2) Substitute the expression from step 1 into the other equation and solve for the other variable. (3) Substitute the value from step 2 into the revised equation from step 1 and solve. Section 7.2 “Solve Linear Systems by Substitution”

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Equation 1 Equation 1 x + 2y = 11 x + 2y = 11 Equation 2 Equation 2 y = 3x + 2 y = 3x + 2 “Solve Linear Systems by Substituting” x + 2y = 11 x + 2y = 11 x + 2(3x + 2) = 11 x + 2(3x + 2) = 11 Substitute x + 6x + 4 = 11 x + 6x + 4 = 11 7x + 4 = 11 7x + 4 = 11 x = 1 x = 1 Equation 1 Equation 1 y = 3x + 2 y = 3x + 2 Substitute value for x into the original equation y = 3(1) + 2 y = 3(1) + 2 y = 5 y = 5 The solution is the point (1,5). Substitute (1,5) into both equations to check. (1) + 2(5) = 11 (1) + 2(5) = 11 11 = 11 (5) = 3(1) + 2 (5) = 3(1) + 2 5 = 5

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Equation 1 Equation 1 4x + 6y = 4 4x + 6y = 4 Equation 2 Equation 2 x – 2y = -6 x – 2y = -6 “Solve Linear Systems by Substituting” 4x + 6y = 4 4x + 6y = 4 4(-6 + 2y) + 6y = 4 4(-6 + 2y) + 6y = 4 Substitute -24 + 8y + 6y = 4 -24 + 8y + 6y = 4 -24 + 14y = 4 -24 + 14y = 4 y = 2 y = 2 Equation 1 Equation 1 x – 2y = -6 x – 2y = -6 Substitute value for x into the original equation x = -6 + 2(2) x = -6 + 2(2) x = -2 x = -2 The solution is the point (-2,2). Substitute (-2,2) into both equations to check. 4(-2) + 6(2) = 4 4(-2) + 6(2) = 4 4 = 4 (-2) - 2(2) = -6 (-2) - 2(2) = -6 -6 = -6 x = -6 + 2y x = -6 + 2y

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Solve a multi-step problem 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. STEP 1 Write a linear system. Let x be the number of pairs of skates rented, and let y be the number of bicycles rented. x + y =25 15x + 30y = 450 Equation for number of rentals Equation for money collected from rentals

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Solve a multi-step problem ANSWER The business rented 20 pairs of skates and 5 bicycles. STEP 2 Solve equation 1 for x. Equation 1 Equation 1 15x + 30y = 450 15x + 30y = 450 Equation 2 Equation 2 x + y = 25 x + y = 25 15x + 30y = 450 15x + 30y = 450 15(25 - y) + 30y = 450 15(25 - y) + 30y = 450 Substitute 375 - 15y + 30y = 450 375 - 15y + 30y = 450 375 + 15y = 450 375 + 15y = 450 y = 5 y = 5 Equation 1 Equation 1 x + y = 25 x + y = 25 Substitute value for x into the original equation x + (5) = 25 x + (5) = 25 x = 20 x = 20 x = 25 - y x = 25 - y 15y = 75 15y = 75

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During a football game, a bag of popcorn sells for $2.50 and a pretzel sells for $2.00. The total amount of money collected during the game was $336. Twice as many bags of popcorn sold compared to pretzels. How many bags of popcorn and pretzels were sold during the game? During a football game, a bag of popcorn sells for $2.50 and a pretzel sells for $2.00. The total amount of money collected during the game was $336. Twice as many bags of popcorn sold compared to pretzels. How many bags of popcorn and pretzels were sold during the game? 96 bags of popcorn and 48 pretzels x = y = y = 2x $2.50y + $2.00x = $336

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Homework Text p. 430, #4-20, 30-34 evens Text p. 430, #4-20, 30-34 evens Text p. 439, #4-24 even, #32, #36 Text p. 439, #4-24 even, #32, #36 NJASK7 prep

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Revision Simultaneous Equations I

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