Solving Systems Using Tables and Graphs€¦ · Solving Systems Using Tables and Graphs. Name Class Date 3-1 Review (continued) The table shows the winning times for the Olympic 400-M

Name Class Date

3-1 Review

As you solve a system of equations, remember the following ideas.

• Lines that have the same slopes but different y-intercepts are parallel and will never intersect. These systems are inconsistent.

• Lines that have both the same slope and the same y-intercept are the same line and will intersect at every point. These systems are dependent.

• Lines that have different slopes will intersect, and the system will have one solution. These systems are independent.

Using a graph or a table, what is the solution of the system of equations?

y = −2x + 8 Write both equations in y = mx + b form.

y = x + 2

y = –2x + 8

Graph the line y = −2x + 8. Graph the line y = x + 2. Circle the point of intersection.

y = x + 2

x = 2, y = 4 Determine the x- and y-coordinates of the point of intersection.

The solution is the ordered pair (2, 4).

Check 2(2) + 4 8 Check by substituting the solution into both equations. 4 + 4 8

8 = 8 4 − 2 2

2 = 2 Exercises Solve each system by graphing or using a table. Check your answers.

1. 2. 3.

4. 5. 6.

7. Which point lies on both Line 1 and Line 2?

(0, 0) (1.875, 1.875)

(2.05, 2.05) (2, 2)

Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

9

Solving Systems Using Tables and Graphs

Name Class Date

3-1 Review (continued)

The table shows the winning times for the Olympic 400-M dash. Use your graphing calculator to find linear models for women’s and men’s winning times. Assuming the trends in the table continue, when will the women’s winning time and the men’s winning time be equal? What will that winning time be?

SOURCE: International Olympic Committee

Step 1 Enter the data into lists on your calculator. L1: number of years since 1968 (value for x) L2: men’s winning times in seconds (value for y1) L3: women’s winning times in seconds (value for y2)

Step 2 Use LinReg(ax + b) to find linear models. This determines the equation of the lines of best f t for the selected data. Use L1 and L2 for the men’s winning times. Use L1 and L3 for the women’s winning times.

Step 3 Graph each model. Use the Intersect feature on the graphing calculator to find the solution of the system. The solution is x = 99.72093 and y = 42.00168.

The linear model shows that if the table’s trends continue, the times for men and women will be equal about 100 years after 1968, in 2068. The winning time will be about 42 seconds.

Exercise 8. The table shows the winning times for Olympic 500-M speed skating. Assuming these trends

continue, when will the women’s winning time equal the men’s winning time? What will that winning time be?



10


Winning Times for the Olympic 400-M Dash (seconds) Year 1968 1972 1976 1980 1984 1988 1992 1996 2000

Men’s Times Women’s Times

43.86 44.66 44.26 44.60 44.27 43.87 43.50 43.49 43.84

52.03 51.08 49.29 48.88 48.83 48.65 48.83 48.25 49.11

Winning Times for the Olympic 500-M Speed Skating (seconds) Year 1968 1972 1976 1980 1984 1988 1992 1994 1998 Men’s Times Women’s Times

40.30 39.44 39.17 38.03 38.19 36.45 37.14 36.33 35.59

46.10 43.33 42.76 41.78 41.02 39.10 40.33 39.25 38.21

Name Class Date

3-2 Review

Follow these steps when solving by substitution.

Step 1 Solve one equation for one of the variables.

Step 2 Substitute the expression for this first variable into the other equation. Solve for the second variable.

Step 3 Substitute the second variable’s value into either equation. Solve for the first variable.

Step 4 Check the solution in the other original equation.

What is the solution of the system of equations? 4 + 3 = 10

+ 2 = 10x yx y

⎧⎨⎩

Step 1 x = −2y + 10 Solve one equation for x.

Step 2 4(−2y + 10) + 3y = 10 Substitute the expression for x into the other equation. −8y + 40 + 3y = 10 Distribute.

–5y = −30 Combine like terms. y = 6 Solve for y.

Step 3 x + 2(6) = 10 Substitute the y value into either equation. x + 12 = 10 Simplify.

x = –2 Solve for x.

Step 4 4(−2) + 3(6) 0 10 Check the solution in the other equation. −8 + 18 10 Simplify. 10 = 10

The solution is (−2, 6).

Exercises Solve each system by substitution.

1. 2. 3. 4.


19

Solving Systems Algebraically

Name Class Date

3-2

Review (continued)

Follow these steps when solving by elimination.

Step 1 Arrange the equations with like terms in columns. Circle the like terms for which you want to obtain coefficients that are opposites.

Step 2 Multiply each term of one or both equations by an appropriate number.

Step 3 Add the equations. Step 4 Solve for the remaining variable.

Step 5 Substitute the value obtained in step 4 into either of the original equations, and solve for the other variable.


What is the solution of the system of equations? 2 – 5 = 113 – 2 = –12x yx y

⎧⎨⎩

Step 1 + 5y = 11 Circle the terms that you want to make opposite. − 2y = −12

Step 2 6x + 15y = 33 Multiply each term of the first equation by 3. −6x + 4y = 24 Multiply each term of the second equation by −2.

Step 3 19y = 57 Add the equations. Step 4 y = 3 Solve for the remaining variable.

Step 5 3x − 2(3) = −12 Substitute 3 for y to solve for x. x = −2

Step 6 2(−2) + 5(3) 11 Check using the other equation. −4 + 15 11

11 = 11

The solution is (−2, 3). You can also check the solution by using a graphing calculator.

Exercises Solve each system by elimination.

5. 6. 7. 8.

9. Reasoning Why does a system with no solution represent parallel lines?


20


Name Class Date

3-3 Review

Solving a System by Using a Table

An English class has 4 computers for at most 18 students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have 4 students and each performance group must have 5 students. In how many ways can you set up the computer groups?

Step 1 Relate the unknowns and define them with variables.

x = number of research groups, y = number of performance groups

number of research groups + number of performance groups ≤ 4

4. number of research groups + 5 number of performance groups ≤ 18

Step 2 Make a table of values for x and y that satisfy the first

inequality. The replacement values for x and y must be whole numbers.

Step 3 In the table, check each pair of values to see which satisfy the other inequality. Highlight these pairs. These are the solutions of the system.

You can have: 0 groups doing research and 0, 1, 2, or 3 groups watching performances or 1 group doing research and 0, 1, or 2 groups watching performances or 2 groups doing research and 0, 1, or 2 groups watching performances or 3 groups doing research and 0 or 1 group watching performances or 4 groups doing research and 0 groups watching performances

Exercises Find the whole number solutions of each system using tables.

1. 2. 3. 4.


29

Systems of Inequalities

x y 0 4, 3, 2, 1, 0

1 3, 2, 1, 0

2 2, 1, 0 3 1, 0

4 0

x + y ≤ 4 4x + 5y ≤ 18

x y 0 4, 3,2, 1, 0

1 3, 2,1, 0

2 2, 1,0

3 1, 0 4 0

Name Class Date

3-3

Review (continued)

Solving a System by Graphing

What is the solution of the system of inequalities?

Step 1 Solve each inequality for y. 2x − y > 1 x + y ≥ 3 −y > −2x + 1 and y ≥ −x + 3 y < 2x − 1

Step 2 Graph the boundary lines. Use a solid line for ≥ or ≤ inequalities. Use a dotted line for > and < inequalities.

Step 3 Shade on the appropriate side of each boundary line. The overlap is the solution to the system.

Exercises Solve each system of inequalities by graphing.

5. y ≤ xy ≥ 3x −1⎧⎨⎩

⎫⎬⎭

6. 2 + > 3

– < 2x yx y

⎧⎨⎩

7. > 1 < + 1xy x

⎧⎨⎩

8. + 3 9

2 – > 1x yx y

≤⎧⎨⎩

9. 1 < – – 13

3 + 1

y x

y x

⎧⎪⎨⎪ ≥⎩

10. 4 + 1

+ 2 –1x yx y

≤⎧⎨

≤⎩

11. 3 > 4 – 3yy x≤⎧

⎨⎩

12. 2 + < 33 – < 2x yx y

⎧⎨⎩

13. –2

3 + 2yx y≥⎧

⎨≤⎩


30

Systems of Inequalities

Name Class Date

3-4 Review

What point in the feasible region maximizes P for the objective function P = 10x + 15y? What point minimizes P?

Step 1 Step 2 Step 3

Graph the constraints and Find the coordinates for Evaluate P at each vertex. shade the feasible region. each vertex of the region.

VERTEX P = 10x + 15y

A (0, 0) P = 10(0) + 15(0) = 0

B (16, 0) P = 10(16) + 15(0) = 160

C (12, 4) P = 10(12) + 15(4) = 180

D (0, 10) P = 10(0) + 15(10) = 150

The maximum value of the objective function is 180. It occurs when x = 12 and y = 4. The minimum value of the objective function is 0. It occurs when x = 0 and y = 0.

Exercises Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.

1. 2. 3.

P = 8x + 2y P = x + 3y P = x − 2y


39

Linear Programming

Name Class Date

3-4

Review (continued)

Your school band is selling calendars as a fundraiser. Wall calendars cost $48 per case of 24. You sell them at $7 per calendar. Pocket calendars cost $30 per case of 40. You sell them at $3 per calendar. You make a profit of $120 per case of wall calendars and $90 per case of pocket calendars. If the band can buy no more than 1000 total calendars and spend no more than $1200, how can you maximize your profit if you sell every calendar? What is the maximum profit?

Relate Organize the information in a table.

Define Let x = number of cases of wall calendars

Let y = number of cases of pocket calendars Write Use the information in the table and the definitions of x and y to write the constraints

and the objective function. Simplify the inequalities if necessary.

24x + 40y ≤ 1000 48x + 30y ≤ 1200 23x + 45y ≤ 125 8x + 5y ≤ 200

Objective function: P = 120x + 90y

Step 1 Step 2 Step 3 Graph the constraints and Find the coordinates for Evaluate the objective function shade to see the feasible region each vertex of the region. using the vertex coordinates.

Linear Programming

Wall Calendars Pocket Calendars Total Number of Cases x y

Number of Units 24x 40y 1000

Cost 48x 30y 1200

Profit 120x 90y 120x + 90y

A(0, 0) P = 120(0) + 90(0) = 0

B(25, 0) P = 120(25) + 90(0) = 3000

C(15, 16) P = 120(15) + 90(16) = 3240

D(0, 25) P = 120(0) + 90(25) = 2250

You can maximize your profit by selling 15 cases of wall

calendars and 16 cases of pocket calendars. The maximum profit is $3240.

Exercises 4. Your band decides to sell the wall calendars for $9 each.

a. How many of each type of calendar should you now buy to maximize your profit?

b. What is the maximum profit?


40

Name Class Date

3-5

Review

What is the solution of the system?

6 2 3 9

2 2 9

x y zx y zx y z

+ + =⎧⎪

− + =⎨⎪− + + =⎩

Use elimination. The equations are numbered to make the process easy to follow.

x + y + z = 6 −x + 2y + 2z = 9

3y + 3z = 15

2x − 2y + 3z = 9 −x + 2y + 2z = 9

2x − y + 3z = 9 −2x + 4y + 4z = 18 3y + 7z = 27 3y + 3z = 15 −3y + (−7z) = −27

−4z = −12 z = 3

3y + 3(3) = 15

3y = 6 y = 2

x + 2 + 3 = 6 x = 1

Pair the equations to eliminate x.

Pair a different set of equations.

Multiply equation 3 by 2 to eliminate x. Then add the two equations.

Equations 4 and 5 form a system. Multiply equation 5 by −1 and add to equation 4 to eliminate y and solve for z.

Substitute z = 3 into equation 4 and solve for y.

Substitute the values of y and z into one of the original equations. Solve for x.

The solution is (1, 2, 3).

Exercises Solve each system by elimination. Check your answers.

1. 2. 3.

4. 5. 6.


49

Systems With Three Variables

Name Class Date

What is the solution of the system? Use substitution.

3 2 19 4 2 3 8

3 2 2 15

x y zx y zx y z

+ − =⎧⎪

− + =⎨⎪− + + =⎩

Step 1 Choose an equation that can be solved easily for

one variable. Choose equation 1 and solve for x. x + 3y − 2z = 19

x = −3y + 2z + 19

Step 2 Substitute the expression for x into equations 2 and 3 and simplify.

4(−3y + 2z + 19) − 2y + 3z = 8 −3(−3y + 2z + 19) + 2y + 2z = 15 −12y + 8z + 76 − 2y + 3z = 8 9y − 6z − 57 + 2y + 2z = 15

−14y + 11z = −68 11y − 4z = 72

−56y + 44z = −272 Multiply by4. 121y − 44z = 792 Multiply by11.

65y = 520 y = 8

−14y + 11z = −68 Substitute y=8 into . –14(8) + 11z = −68

11z = 44 z = 4

Step 4 Use one of the original equations to solve for x.

Review_Systems With Three Variables

Step 3 Write the two new equations as a system. Solve for y and z.

−14y +11z = −6811y − 4z = 72⎧⎨⎩

x + 3y − 2z = 19 Substitute y = 8 and z = 4 into x + 3(8) − 2(4) = 19

x = 3

The solution of the system is (3, 8, 4).

Exercises Solve each system by substitution. Check your answers.

7. 4 3 2 7

2 2 3 152 2 6

x y zx y zx y z

− + − =⎧⎪

− + =⎨⎪− + − = −⎩

8. 2 5 5

3 2 7 102 3 6 12

x y zx y zx y z

− − + = −⎧⎪

+ − =⎨⎪− − + = −⎩

9. 2 3 153 4 2 7 2 2 5 22

x y zx y zx y z

− + =⎧⎪

− − =⎨⎪ + + =⎩

10. 2 2 1

2 2 114 3 2 4

x y zx y zx y z

− + − = −⎧⎪

+ + =⎨⎪ − + =⎩

11. 3 2 7

2 5 4 1 4 6 2

x y zx y zx y z

− + =⎧⎪

− − = −⎨⎪ + − = −⎩

12. 3 3 132 4 5 5

5 2 3

x y zx y zx y z

− − = −⎧⎪

+ − = −⎨⎪− + + =⎩


50

Name Class Date

3-6

Review

How can you represent the system of equations with a matrix?

4x − 3y + 5z = −13x + 3y = 32x + 4y + 3z = 17

⎧

⎨⎪

⎩⎪

Step 1 Write each equation in the same variable order. Line up the like variables. Write in variables that have a coefficient of 0.

4 – 3 5 –13 3 0 3

–2 4 3 17

x y zx y zx y z

+ =⎧⎪

+ + =⎨⎪ + + =⎩

Step 2 Write the matrix using the coefficients and constants. Remember to enter a 1 for variables with no numeric coefficient.

4 3 5 –131 3 0 32 4 3 17

⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

Exercises Write a matrix to represent each system.

1. 2 –33 5x yy+ =⎧

⎨=⎩

2. 3 – 5 2 94 7 32 – 12

x y zx y zx z

+ =⎧⎪

+ + =⎨⎪ =⎩

3. 5 – 3 23 2 64 3 1

x y zy zx y z

+ =⎧⎪

+ =⎨⎪ + + =⎩

4. 2 – 35 4 –5– 2 1

x zy zx y

=⎧⎪

+ =⎨⎪ + =⎩

5. 6

23 – 2 – 5 10

zx yx y z

=⎧⎪

+ =⎨⎪ =⎩

6. 2 – 5 3 4– 2 4 –23 – 2 –5

z x yy x zx z y

+ =⎧⎪

+ + =⎨⎪ + =⎩


59

Solving Systems Using Matrices

Name Class Date

3-6

Review (continued)

What is the solution of the system? 2 5 5– 2 –7x yx y+ =⎧

⎨+ =⎩

Step 1 Write the matrix for the system.

2 5 5

1 2 –7⎡ ⎤⎢ ⎥−⎣ ⎦

Step 2 Multiply Row 2 by 2. Add to Row 1. Replace Row 1 with the sum. Write the new matrix.

2 5 5

+ 2(–1 2 –7)

0 9 –9

Step 3 Divide Row 1 by 9. Write the new matrix.

Solving Systems Using Matrices

0 9 –9

1 2 –7⎡ ⎤⎢ ⎥−⎣ ⎦

0 1 –11 (0 9 –9) 1 2 –79

⎡ ⎤⎢ ⎥−⎣ ⎦

Step 4 Multiply Row 1 by –2. Add to Row 2. Replace Row 2 with the sum. Write the new matrix.

–2(0 1 –1)

+ −1 2 –7

–1 0 –5

Step 5 Multiply Row 2 by 21. Write the new matrix.

−1 −1 0 −5( )

0 11 0

–15

⎡

⎣⎢⎢

⎤

⎦⎥⎥

This matrix is equivalent to the system –1 5yx=⎧

⎨=⎩

. The solution is (5, −1).

Exercises Solve each system of equations using a matrix.

7. 4 3 6– – –1x yx y+ =⎧

⎨=⎩

8. 6 –2– 3 13x yx y+ =⎧

⎨+ =⎩

9. 3 2 –4–4 – 3 7x yx y+ =⎧

⎨=⎩


60

0 1 –1

1 0 –5⎡ ⎤⎢ ⎥−⎣ ⎦

Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

9

Name Class Date

As you solve a system of equations, remember the following ideas.

• Lines that have the same slopes but diff erent y-intercepts are parallel and will never intersect. Th ese systems are inconsistent.

• Lines that have both the same slope and the same y-intercept are the same line and will intersect at every point. Th ese systems are dependent.

• Lines that have diff erent slopes will intersect, and the system will have one solution. Th ese systems are independent.

Problem

Using a graph or a table, what is the solution of the system of equations? e 2x 1 y 5 8

y 2 x 5 2

y 5 22x 1 8 Write both equations in y 5 mx 1 b form.

y 5 x 1 2

Graph the line y 5 22x 1 8. Graph the line y 5 x 1 2. Circle the point of intersection.

x 5 2, y 5 4 Determine the x- and y -coordinates of the point of intersection.

Th e solution is the ordered pair (2, 4).

Check 2(2) 1 4 0 8 Check by substituting the solution into both equations.

4 1 4 0 8 8 5 8 �

4 2 2 0 22 5 2 �

Exercises

Solve each system by graphing or using a table. Check your answers.

1. e 3x 1 y 5 6

3x 1 y 5 3 2. e22x 1 y 1 3 5 0

x 2 1 5 y 3. e x 1 y 5 3

y 5 3x 2 1

4. e y 5 1 2 x

2x 1 y 5 4 5. e2x 1 2y 5 2

3x 1 2y 5 26 6. e2x 1 y 5 22

22x 1 3y 5 23

7. Which point lies on both Line 1 and Line 2?

(0, 0) (1.875, 1.875)

(2.05, 2.05) (2, 2)

y � �2x � 8

y � x � 2 x

y

O2 4 6 8�2

468

3-1 Reteaching Solving Systems Using Tables and Graphs

xO 2Line 1

4

Line 24

�4

y

(1, 3)

(3, 22)

(2, 1)

(22, 0)

C

(1, 2)

(3, 1)


10

Name Class Date

Problem

Th e table shows the winning times for the Olympic 400-M dash. Use your graphing calculator to fi nd linear models for women’s and men’s winning times. Assuming the trends in the table continue, when will the women’s winning time and the men’s winning time be equal? What will that winning time be?

Step 1 Enter the data into lists on your calculator.

L1: number of years since 1968 (value for x)

L2: men’s winning times in seconds (value for y1)

L3: women’s winning times in seconds (value for y2)

Step 2 Use LinReg(ax 1 b) to fi nd linear models. Th is determines the equation of the lines of best fi t for the selected data.

Use L1 and L2 for the men’s winning times.

Use L1 and L3 for the women’s winning times.

Step 3 Graph each model. Use the Intersect feature on the graphing calculator to fi nd the solution of the system. Th e solution is x 5 99.72093 and y 5 42.00168.

Th e linear model shows that if the table’s trends continue, the times for men and women will be equal about 100 years after 1968, in 2068. Th e winning time will be about 42 seconds.

Exercise 8. Th e table shows the winning times for Olympic 500-M speed skating.

Assuming these trends continue, when will the women’s winning time equal the men’s winning time? What will that winning time be?

Winning Times for the Olympic 400-M Dash (seconds)

Year 1968

52.03

43.86

51.08

1972

44.66

49.29

1976

44.26

1980

48.88

44.60

1984

48.83

44.27

1988

48.65

43.87

1992

48.83

43.50

1996

48.25

43.49

2000

49.11

43.84Men’sTimes

Women’sTimes


�

��

Intersectionx � 99.72093 y � 42.00168

Winning Times for the Olympic 500-M Speed Skating (seconds)

Year 1968

46.10

40.30

43.33

1972

39.44

42.76

1976

39.17

1980

41.78

38.03

1984

41.02

38.19

1988

39.10

36.45

1992

40.33

37.14

1994

39.25

36.33

1998

38.21

35.59Men’sTimes

Women’sTimes


3-1 Reteaching (continued)


2028; 31.265 seconds


19

Name Class Date

Follow these steps when solving by substitution.

Step 1 Solve one equation for one of the variables.

Step 2 Substitute the expression for this fi rst variable into the other equation. Solve for the second variable.

Step 3 Substitute the second variable’s value into either equation. Solve for the fi rst variable.


Problem

What is the solution of the system of equations? e 4x 1 3y 5 10

4x 1 2y 5 10

Step 1 x 5 22y 1 10 Solve one equation for x.

Step 2 4(22y 1 10) 1 3y 5 10 Substitute the expression for x into the other equation. 28y 1 40 1 3y 5 10 Distribute. 25y 5 230 Combine like terms. y 5 6 Solve for y.

Step 3 x 1 2(6) 5 10 Substitute the y value into either equation. x 1 12 5 10 Simplify. x 5 22 Solve for x.

Step 4 4(22) 1 3(6) 0 10 Check the solution in the other equation. 28 1 18 0 10 Simplify. 10 5 10 �

Th e solution is (22, 6).

Exercises

Solve each system by substitution.

1. e2x 2 3y 5 2

2x 1 2y 5 5 2. ea 1 3b 5 4

a 5 22 3. e22m 1 n 5 6

27m 1 6n 5 1 4. e 7x 2 3y 5 21

x 1 2y 5 12

3-2 ReteachingSolving Systems Algebraically

x 5 219, y 5 27 a 5 22, b 5 2 m 5 27, n 5 28 x 5 2, y 5 5


20

Name Class Date

Follow these steps when solving by elimination.

Step 1 Arrange the equations with like terms in columns. Circle the like terms for which you want to obtain coeffi cients that are opposites.

Step 2 Multiply each term of one or both equations by an appropriate number.

Step 3 Add the equations.

Step 4 Solve for the remaining variable.

Step 5 Substitute the value obtained in step 4 into either of the original equations, and solve for the other variable.


Problem

What is the solution of the system of equations? e 2x 1 5y 5 211

3x 2 2y 5 212

Step 1 2x 1 5y 5 211 Circle the terms that you want to make opposite. 3x 2 2y 5 212

Step 2 6x 1 15y 5 33 Multiply each term of the fi rst equation by 3. 26x 1 14y 5 24 Multiply each term of the second equation by 22.

Step 3 19y 5 57 Add the equations.Step 4 y 5 3 Solve for the remaining variable.

Step 5 3x 2 2(3) 5 212 Substitute 3 for y to solve for x. x 5 22

Step 6 2(22) 1 5(3) 0 11 Check using the other equation. 24 1 15 0 11 11 5 11 �

Th e solution is (22, 3). You can also check the solution by using a graphing calculator.

Exercises

Solve each system by elimination.

5. e 3x 1 2y 5 217

3x 2 3y 5 219 6. e25f 1 4m 5 26

22f 2 3m 5 21 7. e23x 2 2y 5 5

26x 1 4y 5 7 8. e22x 2 24y 5 212

10x 1 20y 5 210

9. Reasoning Why does a system with no solution represent parallel lines?



If there is no solution, then there are no values of the variables that will make both equations true. This means there is no point that lies on both lines, so the lines never meet and are therefore parallel.

x 5 23, y 5 24 f 5 2, m 5 21 no solution y 5 212x 2 12, where

x is any real number


29

Name Class Date

Solving a System by Using a Table

Problem

An English class has 4 computers for at most 18 students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have 4 students and each performance group must have 5 students. In how many ways can you set up the computer groups?

Step 1 Relate the unknowns and defi ne them with variables.

x 5 number of research groups, y 5 number of performance groups number of research groups 1 number of performance groups # 4

4 ? number of research groups 1 5 ? number of performance groups # 18

Step 2 Make a table of values for x and y that satisfy the fi rst inequality. Th e replacement values for x and y must be whole numbers.

Step 3 In the table, check each pair of values to see which satisfy the other inequality. Highlight these pairs. Th ese are the solutions of the system.

You can have:0 groups doing research and 0, 1, 2, or 3 groups watching performances or1 group doing research and 0, 1, or 2 groups watching performances or 2 groups doing research and 0, 1, or 2 groups watching performances or3 groups doing research and 0 or 1 group watching performances or4 groups doing research and 0 groups watching performances

Exercises

Find the whole number solutions of each system using tables.

1. e x 1 y , 4

x 1 2y # 10 2. e 6x 2 3y $ 1

6x 1 3y # 21 3. e x 1 y $ 5

y , 22x 1 8 4. e y , 3

4x 1 2y , 12

x 1 y # 44x 1 5y # 18

3-3 ReteachingSystems of Inequalities

x

0 4, 3, 2, 1, 0

1 3, 2, 1, 0

2 2, 1, 0

3 1, 0

4 0

y

x

0 4, 3, 2, 1, 0

1 3, 2, 1, 0

2 2, 1, 0

3 1, 0

4 0

y

(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (3, 0)

(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)

(0, 5), (0, 6), (0, 7), (1, 4), (1, 5), (2, 3)

(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)


30

Name Class Date

Solving a System by Graphing

Problem

What is the solution of the system of inequalities? e 2x 2 y . 1

2x 1 y $ 3Step 1 Solve each inequality for y. 2x 2 y . 1 x 1 y $ 3

2y . 22x 1 1 and y $ 2x 1 3y , 2x 2 1

Step 2 Graph the boundary lines. Use a solid line for $ or #

xO 2 4�4 �2

�2

2

4

�4

y

inequalities. Use a dotted line for . and , inequalities.

Step 3 Shade on the appropriate side of each boundary line.

xO 2 4�4 �2

�2

2

4

�4

y

Th e overlap is the solution to the system.

Exercises

Solve each system of inequalities by graphing.

5. e y # xy $ 3x 2 1

6. e 2x 1 y . 3

2x 2 y , 2 7. e x . 1

y , x 1 1

8. e 2x 1 3y # 9

2x 2 3y . 1 9. e y , 2

13x 2 1

y $ 3x 1 1 10. e 4x 1 2y # 1

4x 1 2y # 21

11. e y # 3

y . 4x 2 3 12. e 2x 1 y , 3

3x 2 y , 2 13. e y $ 22

3x 1 y # 2

3-3 Reteaching (continued) Systems of Inequalities

O

2

�2�2

2x

y

O

2

�2�2

2

x

y

O

2

�2�2

2

x

y

O

2

�2�2

2x

y

O

2

�2�2

2 x

y

O2

�2�2

2

x

y

O

2

�2�2

2x

y

O

2

�2�2

2

x

y

O

2

�2�2

2

x

y


39

Name Class Date

Problem

What point in the feasible region maximizes P for the objective function P 5 10x 1 15y? What point minimizes P?

Constraints dx 1 y # 16

3x 1 6y # 60

x $ 0

y $ 0

Step 1 Step 2 Step 3

Graph the constraints and shade the feasible region.

Find the coordinates for each vertex of the region.

Evaluate P at each vertex.

VERTEX P 5 10x 1 15y

A (0, 0) P 5 10(0) 1 15(0) 5 0

B (16, 0) P 5 10(16) 1 15(0) 5 160

C (12, 4) P 5 10(12) 1 15(4) 5 180

D (0, 10) P 5 10(0) 1 15(10) 5 150

Th e maximum value of the objective function is 180. It occurs when x 5 12 and y 5 4.

Th e minimum value of the objective function is 0. It occurs when x 5 0 and y 5 0.

Exercises

Graph each system of constraints. Name all vertices. Th en fi nd the values of x and y that maximize or minimize the objective function.

1. d5y 1 4x # 35

5y 1 x $ 20

y # 6

x $ 1

2. dx 1 y $ 2

x $ yx # 4

y $ 0

3. c3x 1 4y $ 12

5x 1 6y # 30

1 # x # 3

P 5 8x 1 2y P 5 x 1 3y P 5 x 2 2y

3-4 ReteachingLinear Programming

xO

y

D

C

BA

8

8

16

vertices: (1, 6), Q1, 195 R ,

Q54, 6R , (5, 3)

max: 46 at (5, 3);

min: 785 at Q1, 19

5 R

vertices: (4, 0), (1, 1), (2, 0), (4, 4)

max: 16 at (4, 4); min: 2 at (2, 0)

vertices: Q1, 94R , Q1, 25

6 R , Q3, 3

4R , Q3, 52R

y

x2

O

4

6

4 62

(5, 3)

(1, 6)(5/4, 6)

(1, 19/5)

y

x2

O

4

6

4 62(2, 0)

(4, 4)

(4, 0)

(1, 1) y

x2

O

4

6

4 62

(1, 9/4)

(1, 3/6)

(3, 5/2)

max: 32 at Q3, 34R ;

min: 2 223 at Q1, 25

6 R


40

Name Class Date

Problem

Your school band is selling calendars as a fundraiser. Wall calendars cost $48 per case of 24. You sell them at $7 per calendar. Pocket calendars cost $30 per case of 40. You sell them at $3 per calendar. You make a profi t of $120 per case of wall calendars and $90 per case of pocket calendars. If the band can buy no more than 1000 total calendars and spend no more than $1200, how can you maximize your profi t if you sell every calendar? What is the maximum profi t?

Relate Organize the information in a table.

Defi ne Let x 5 number of cases of wall calendars Let y 5 number of cases of pocket calendarsWrite Use the information in the table and the defi nitions of x and y to write the constraints

and the objective function. Simplify the inequalities if necessary. 24x 1 40y # 1000 48x 1 30y # 1200 23x 1 45y # 125 8x 1 5y # 200

c3x 1 5y # 125

8x 1 5y # 200

x $ 0, y $ 0

Objective function: P 5 120x 1 90y

Step 1 Step 2 Step 3Graph the constraints and shade to see the feasible region.

Find the coordinates for each vertex of the region.

Evaluate the objective function using the vertex coordinates.

A(0, 0) P 5 120(0) 1 90(0) 5 0

B(25, 0) P 5 120(25) 1 90(0) 5 3000

C(15, 16) P 5 120(15) 1 90(16) 5 3240

D(0, 25) P 5 120(0) 1 90(25) 5 2250

You can maximize your profi t by selling 15 cases of wall calendars and 16 cases of pocket calendars. Th e maximum profi t is $3240.

Exercises 4. Your band decides to sell the wall calendars for $9 each. a. How many of each type of calendar should you now buy to maximize your

profi t? b. What is the maximum profi t?

4

48

12162024

8 12 16 20 24

y

xO


Linear Programming

Number of Cases

Number of Units

Cost

Profit

Wall Calendars

x

24x

48x

120x

Pocket Calendars

y

40y

30y

90y

Total

1000

1200

120x 1 90y

25 cases of wall calendars and no cases of pocket calendars$4200


49

Name Class Date

Problem

What is the solution of the system?

•x 1 2y 1 3z 5 216

2x 2 2y 1 3z 5 219

2x 1 2y 1 2z 5 219

Use elimination. Th e equations are numbered to make the process easy to follow.

x 1 2y 1 2z 5 216

2x 1 2y 1 2z 5 219

3y 1 3z 5 215

2x 2 2y 1 3z 5 19

2x 1 2y 1 2z 5 29

2x 2 4y 1 3z 5 219

22x 1 4y 1 4z 5 218

3y 1 7z 5 227

3y 1 (23z) 5 215

23y 1 (27z) 5 227

24z) 5 212

z) 5 213

3y 1 3(3) 5 215

3y 5 216

y 5 212

x 1 2 1 3 5 216

x 5 211

Th e solution is (1, 2, 3).

ExercisesSolve each system by elimination. Check your answers.

1. •2x 2 3y 1 2z 5 10

4x 1 2y 2 5z 5 10

4x 2 3y 1 5z 5 8

2. •3x 2 3y 1 2z 5 6

2x 1 3y 1 2z 5 2

3x 1 5y 1 4z 5 4

3. •6x 2 4y 1 5z 5 31

5x 1 2y 1 2z 5 13

5x 1 4y 1 5z 5 2

4. •3x 1 2y 1 3z 5 2

4x 2 2y 1 3z 5 24

2x 1 2y 1 2z 5 8

5. •5x 1 2y 1 3z 5 5

3x 2 3y 2 3z 5 9

3x 1 2y 1 4z 5 6

6. •4x 1 3y 1 2z 5 21

4x 1 3y 1 2z 5 210

2x 2 4y 2 2z 5 26

A

B

C

A

C

Pair the equations to eliminate x .

B

C

D

B

C

E

Multiply equation 3 by 2 to eliminate x . Then add the two equations.

D

E

Equations 4 and 5 form a system.

Multiply equation 5 by 21 and add to equation 4 to eliminate y and solve for z.

A

D Substitute z 5 3 into equation 4 and solve for y .

Substitute the values of y and z into one of the original equations. Solve for x .

3-5 ReteachingSystems With Three Variables

Pair a different set of equations.

(23, 22, 4)

(4, 2, 2) (2, 22, 2) (3, 22, 1)

(21, 3, 2) (2, 24, 3)


50

Name Class Date

Problem

What is the solution of the system? Use substitution. •24x 1 3y 2 2z 5 19

24x 2 2y 1 3z 5 8

23x 1 2y 1 2z 5 15

Step 1 Choose an equation that can be solved easily for one variable. Choose equation 1 and solve for x.

Step 2 Substitute the expression for x into equations 2 and 3 and simplify.

4(23y 1 2z 1 19) 2 2y 1 3z 5 8 23(23y 1 2z 1 19) 1 2y 1 2z 5 15212y 1 8z 1 76 2 2y 1 3z 5 8 9y 2 6z 2 57 1 2y 1 2z 5 15

D 214y 1 11z 5 268 E 11y 2 4z 5 72

Step 3 Write the two new equations as a system. Solve for y and z. e214y 1 11z 5 268

211y 2 14z 5 272

256y 1 44z 5 2272 Multiply D by 4. 214y 1 11z 5 268 Substitute y 5 8 into D.121y 2 44z 5 2792 Multiply E by 11. 214(8) 1 11z 5 268

65y 5 520 11z 5 44y 5 8 z 5 4

Step 4 Use one of the original equations to solve for x.x 1 3y 2 2z 5 19 Substitute y 5 8 and z 5 4 into A.

x 1 3(8) 2 2(4) 5 19x 5 3

Th e solution of the system is (3, 8, 4).

Exercises

Solve each system by substitution. Check your answers.

7. •24x 1 3y 2 2z 5 7

2x 2 2y 1 3z 5 15

2x 1 2y 2 2z 5 26

8. •22x 2 2y 1 5z 5 25

23x 1 2y 2 7z 5 10

22x 2 3y 1 6z 5 212

9. •2x 2 4y 1 3z 5 15

3x 2 4y 2 2z 5 7

2x 1 2y 1 5z 5 22

10. •22x 1 2y 2 2z 5 21

22x 1 2y 1 2z 5 11

24x 2 3y 1 2z 5 4

11. •2x 2 3y 1 2z 5 7

2x 2 5y 2 4z 5 21

2x 1 4y 2 6z 5 22

12. •23x 2 4y 2 3z 5 213

22x 1 4y 2 5z 5 25

25x 1 2y 1 5z 5 3

A

B

C

A x 1 3y 2 2z 5 19

x 5 23y 1 2z 1 19

D

E


Systems With Three Variables

(22, 7, 11)

(1, 2, 3)

(3, 4, 1)

(6, 1, 2)

(5, 1, 2)

(2, 4, 5)


59

Name Class Date

Problem

How can you represent the system of equations with a matrix? c24x 2 3y 1 5z 5 213

24x 1 3y 5 3

22x 1 4y 1 3z 5 17

Step 1 Write each equation in the same variable order. Line up the like variables. Write in variables that have a coeffi cient of 0.

c24x 2 3y 1 5z 5 213

24x 1 3y 1 0z 5 213

22x 1 4y 1 3z 5 217

Step 2 Write the matrix using the coeffi cients and constants. Remember to enter a 1 for variables with no numeric coeffi cient.

D 4 23 5

1 3 0

22 4 3

4 213

3

17

TExercises

Write a matrix to represent each system.

1. e 2x 1 y 5 23

3y 5 5 2. c3x 2 5y 1 2z 5 9

4x 1 7y 1 z 5 3

2x 2 z 5 12

3. c5x 2 y 1 3z 5 2

3y 1 2z 5 6

4x 1 3y 1 z 5 1

4. c2x 2 z 5 3

5y 1 4z 5 25

2x 1 2y 5 1

5. cz 5 6

x 1 y 5 2

3x 2 2y 2 5z 5 10

6. c2z 2 5x 1 3y 5 4

2y 1 2x 1 4z 5 22

3x 1 z 2 2y 5 25

3-6 ReteachingSolving Systems Using Matrices

c2 10 3

` 235d

D 2 0 210 5 4

21 2 0 4 3

251T

D3 25 24 7 12 0 21

4 93

12T

D0 0 11 1 03 22 25

4 62

10T

D5 21 30 3 24 3 1

4 261T

D25 3 22 21 43 22 1

4 42225T


60

Name Class Date

Problem

What is the solution of the system? e 2x 1 5y 5 5

2x 1 2y 5 27

Step 1 Write the matrix for the system.

c 2 5

21 2 ` 5

27d


2 5 5)

12(21 2 27)

0 9 29)

c 0 9

21 2 ` 29

27d

Step 3 Divide Row 1 by 9. Write the new matrix.

19(0 9 29) c 0 1

21 2 ` 21

27d


22(0 1 21)

121 2 27)

21 0 25)

c 0 1

21 0 ` 21

25d

Step 5 Multiply Row 2 by 21. Write the new matrix.

21(21 0 25) c0 1

1 0 ` 21

5d

Th is matrix is equivalent to the system e y 5 21

x 5 5. Th e solution is (5, 21).

Exercises

Solve each system of equations using a matrix.

7. e24x 1 3y 5 6

2x 2 y 5 21 8. e 6x 1 y 5 22

2x 1 3y 5 13 9. e 3x 1 2y 5 24

24x 2 3y 5 7

3-6 Reteaching (continued) Solving Systems Using Matrices

(21, 4) (2, 25)(3, 22)

Documents

Solving Systems Using Tables and Graphs€¦ · Solving Systems Using Tables and Graphs. Name Class Date 3-1 Review (continued) The table shows the winning times for the Olympic 400-M