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Copy HW in your planner.Copy HW in your planner. Text p. 430, #4-20 evens, 30-34 evensText p. 430, #4-20 evens, 30-34 evens Text p. 439, #4-24 even, #32, #36Text p. 439, #4-24 even, #32, #36

Open your textbook to page 424 and Open your textbook to page 424 and preview Chapter 7 “Systems of preview Chapter 7 “Systems of Equations and Inequalities”Equations and Inequalities”

Chapter 7 Preview Chapter 7 Preview “Solving and Graphing Linear “Solving and Graphing Linear

Systems”Systems”(7.1) Solve Linear Systems by Graphing(7.1) Solve Linear Systems by Graphing

(7.2) Solve Linear Systems by Substituting(7.2) Solve Linear Systems by Substituting

(7.3) Solve Linear Systems by Adding or Subtracting(7.3) Solve Linear Systems by Adding or Subtracting

(7.4) Solve Linear Systems by Multiplying First(7.4) Solve Linear Systems by Multiplying First

(7.5) Solve Special Types of Linear Systems(7.5) Solve Special Types of Linear Systems

(7.6) Solve Systems of Linear Inequalities(7.6) Solve Systems of Linear Inequalities

Linear SystemLinear System– – consists of two or more linear equations. consists of two or more linear equations.

Equation 1 Equation 1

3x – 2y = 53x – 2y = 5 Equation 2Equation 2

x + 2y = 7 x + 2y = 7

Section 7.1Section 7.1“Solve Linear Systems by Graphing”“Solve Linear Systems by Graphing”

A solution to a linear system is an orderedA solution to a linear system is an ordered

pair (a point) where the two linear equationspair (a point) where the two linear equations

(lines) intersect (cross). (lines) intersect (cross).

Solving a Linear System by Solving a Linear System by GraphingGraphing

(1) (1) GraphGraph both equations in the same plane. both equations in the same plane.

(2) (2) EstimateEstimate the coordinates of the point the coordinates of the point where the two lines intersect.where the two lines intersect.

(3) (3) Check the coordinate by substituting Check the coordinate by substituting into EACH equation of the linear systeminto EACH equation of the linear system, to , to see if the point is a solution for both see if the point is a solution for both equations.equations.

7 = 77 = 7

SOLUTIONSOLUTION

Use the graph to solve the Use the graph to solve the system. Then check your system. Then check your solution algebraicallysolution algebraically..

x x + 2+ 2y y = 7= 7 Equation Equation 11

33x x – 2– 2y y = 5= 5 EquationEquation 22

The lines appear to intersect at the point The lines appear to intersect at the point (3, 2).(3, 2).

CHECKCHECK SubstituteSubstitute 33 forfor xx andand 22 forfor yy in each equationin each equation..

xx ++ 22yy == 77

33 + + 2(2(22)) ==??

77

33xx –– 22yy == 55

5 = 55 = 53(3(33)) –– 2(2(22)) 55==

??

Equation Equation 11 EquationEquation 22 Because the Because the ordered pair (3, 2) ordered pair (3, 2) is a solution of is a solution of each equation, it is each equation, it is a solution of the a solution of the system.system.

Using a Graph to Solve a Linear Using a Graph to Solve a Linear SystemSystem

SOLUTIONSOLUTION

Use the graph to solve the Use the graph to solve the system. Then check your system. Then check your solution algebraicallysolution algebraically..

x x + 4+ 4y y = -8= -8 Equation Equation 11

-x +-x + y y = -7= -7 EquationEquation 22

The lines appear to intersect at the point The lines appear to intersect at the point (4, -3).(4, -3).

CHECKCHECK SubstituteSubstitute 4 4 forfor xx andand -3 -3 forfor yy in each equationin each equation..

xx ++ 44yy == -8-8

44 + + 4(4(-3-3))==??

-8-8

-x-x ++ yy == -7-7

-7= -7-7= -7 -4-4 ++ ((-3-3)) -7-7==

??

Equation Equation 11 EquationEquation 22 Because the Because the ordered pair (4, -3) ordered pair (4, -3) is a solution of is a solution of each equation, it is each equation, it is a solution of the a solution of the system.system.

Using a Graph to Solve a Linear Using a Graph to Solve a Linear SystemSystem

-8 = -8-8 = -8

Standardized Test PracticeStandardized Test Practice

Which system of equations can be used to find the number Which system of equations can be used to find the number xx of sessions of tennis after which the total cost of sessions of tennis after which the total cost yy with a season with a season pass, including the cost of the pass, is the same as the total pass, including the cost of the pass, is the same as the total cost without a season pass?cost without a season pass?

y y = 13= 13xx y y = 4= 4xxAA

y y = 13= 13xx y y = 90 + 4= 90 + 4xx

CC

y y = 4= 4xxy y = 90 + 13= 90 + 13xx

BB

y y = 90 + 4= 90 + 4xxy y = 90 + 13= 90 + 13xx

DD

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.

Standardized Test PracticeStandardized Test PracticeWhich system of equations can be used to find the number Which system of equations can be used to find the number xx of sessions of tennis after which the total cost of sessions of tennis after which the total cost yy with a season with a season pass, including the cost of the pass, is the same as the total pass, including the cost of the pass, is the same as the total cost without a season pass?cost without a season pass?

y y = 13= 13xx y y = 4= 4xxAA

y y = 13= 13xx y y = 90 + 4= 90 + 4xx

CC

y y = 4= 4xxy y = 90 + 13= 90 + 13xx

BB

y y = 90 + 4= 90 + 4xxy y = 90 + 13= 90 + 13xx

DD

EQUATION 1

y = 13 x

EQUATION 2

y = 90 + 4 x

Solve a multi-step problemSolve a multi-step problem

A business rents in-line skates and A business rents in-line skates and bicycles. During one day, the business bicycles. During one day, the business has a total of has a total of 2525 rentals and collects rentals and collects $450$450 for the rentals. Find the number of for the rentals. Find the number of pairs of skates rented and the number pairs of skates rented and the number of bicycles rented.of bicycles rented.

STEPSTEP 11Write a linear system. Let Write a linear system. Let xx be the number of pairs be the number of pairs of skates rented, and let of skates rented, and let yy be the number of be the number of bicycles rented. bicycles rented.

x x + + y y =25=25

1515x x + 30+ 30y y = 450= 450

Equation for number of rentalsEquation for number of rentals

Equation for money collected from rentalsEquation for money collected from rentals

Solve a multi-step problemSolve a multi-step problem

STEPSTEP 33

Estimate the point of intersectionEstimate the point of intersection.. The two lines appear to intersect atThe two lines appear to intersect at (20, 5)(20, 5)..

STEPSTEP 44Check whether Check whether (20, 5)(20, 5) is a solution is a solution..

2020 ++ 55 2525==?? 15(15(2020)) ++ 30(30(55) 450) 450==??

450 450 == 45045025 25 == 2525

ANSWERANSWER

The business rented The business rented 20 20 pairs of skates and pairs of skates and 55 bicycles. bicycles.

STEPSTEP 22Graph both equationsGraph both equations..

Solving a Linear System by SubstitutionSolving a Linear System by Substitution

(1) (1) SolveSolve one of the equations for one of its variables. one of the equations for one of its variables. (When possible, solve for a variable that has a coefficient (When possible, solve for a variable that has a coefficient of 1 or -1).of 1 or -1).

(2) (2) SubstituteSubstitute the expression from step 1 into the other the expression from step 1 into the other equation and solve for the other variable.equation and solve for the other variable.

(3) (3) SubstituteSubstitute the value from step 2 into the revised the value from step 2 into the revised equation from step 1 and solve.equation from step 1 and solve.

Section 7.2Section 7.2“Solve Linear Systems by “Solve Linear Systems by

Substitution”Substitution”

Equation 1 Equation 1

x + 2y = 11x + 2y = 11 Equation 2Equation 2

y = 3x + 2 y = 3x + 2

““Solve Linear Systems by Solve Linear Systems by Substituting”Substituting”

x + 2y = 11x + 2y = 11 x + 2x + 2(3x + 2)(3x + 2) = 11 = 11 Substitute Substitute

x + x + 6x + 46x + 4 = 11 = 11 7x 7x + 4+ 4 = 11 = 11

x = 1x = 1

Equation 1 Equation 1 y = 3x + 2 y = 3x + 2 Substitute value forSubstitute value forx into the original x into the original equation equation y = 3(1) + 2y = 3(1) + 2

y = 5y = 5

The solution is the point (1,5). The solution is the point (1,5). Substitute (1,5) into both Substitute (1,5) into both

equations to check.equations to check.

(1) + 2(5) = 11(1) + 2(5) = 1111 = 1111 = 11

(5) = 3(1) + 2(5) = 3(1) + 25 = 5 5 = 5

Equation 1 Equation 1

4x + 6y = 44x + 6y = 4 Equation 2Equation 2

x – 2y = -6 x – 2y = -6

““Solve Linear Systems by Solve Linear Systems by Substituting”Substituting”

4x + 6y = 44x + 6y = 4 44(-6 + 2y)(-6 + 2y) + 6y = 4 + 6y = 4 Substitute Substitute

-24 + 8y-24 + 8y + 6y + 6y = 4 = 4 -24 -24 + 14y+ 14y = 4 = 4

y = 2y = 2

Equation 1 Equation 1 x – 2y = -6 x – 2y = -6 Substitute value forSubstitute value forx into the original x into the original equation equation x = -6 + 2(2)x = -6 + 2(2)

x = -2x = -2

The solution is the point (-2,2). The solution is the point (-2,2). Substitute (-2,2) into both Substitute (-2,2) into both

equations to check.equations to check.

4(-2) + 6(2) = 44(-2) + 6(2) = 44 = 44 = 4

(-2) - 2(2) = -6(-2) - 2(2) = -6-6 = -6 -6 = -6

x = -6 + 2y x = -6 + 2y

Solve a multi-step problemSolve a multi-step problem

A business rents in-line skates and A business rents in-line skates and bicycles. During one day, the business bicycles. During one day, the business has a total of has a total of 2525 rentals and collects rentals and collects $450$450 for the rentals. Find the number of for the rentals. Find the number of pairs of skates rented and the number pairs of skates rented and the number of bicycles rented.of bicycles rented.

STEPSTEP 11Write a linear system. Let Write a linear system. Let xx be the number of pairs be the number of pairs of skates rented, and let of skates rented, and let yy be the number of be the number of bicycles rented. bicycles rented.

x x + + y y =25=25

1515x x + 30+ 30y y = 450= 450

Equation for number of rentalsEquation for number of rentals

Equation for money collected from rentalsEquation for money collected from rentals

Solve a multi-step problemSolve a multi-step problem

ANSWERANSWER

The business rented The business rented 20 20 pairs of skates and pairs of skates and 55 bicycles. bicycles.

STEPSTEP 22Solve equation 1 for x.Solve equation 1 for x.

Equation 1 Equation 1

15x + 30y = 45015x + 30y = 450 Equation 2Equation 2

x + y = 25 x + y = 25

15x + 30y = 45015x + 30y = 450 1515(25 - y)(25 - y) + 30y = 450 + 30y = 450Substitute Substitute

375 - 15y375 - 15y + 30y + 30y = 450 = 450

375 375 + 15y+ 15y = 450 = 450

y = 5y = 5

Equation 1 Equation 1 x + y = 25 x + y = 25 Substitute value forSubstitute value forx into the original x into the original equation equation x + (5) = 25x + (5) = 25

x = 20x = 20

x = 25 - y x = 25 - y

15y15y = 75 = 75

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

96 bags of popcorn and 48 pretzels96 bags of popcorn and 48 pretzels

x = x =

y = y =

y = 2xy = 2x

$2.50y + $2.00x = $336$2.50y + $2.00x = $336

HomeworkHomework Text p. 430, #4-20, 30-34 evensText p. 430, #4-20, 30-34 evens Text p. 439, #4-24 even, #32, #36Text p. 439, #4-24 even, #32, #36

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