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 Solving Systems Using Tables and Graphs
27
Embed
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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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.
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?
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.
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.
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.
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?
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
SOURCE: International Olympic Committee
�
�������
Intersectionx � 99.72093 y � 42.00168
Winning Times for the Olympic 500-M Speed Skating (seconds)
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.
Step 4 Check the solution in the other original equation.
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
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.
Step 6 Check the solution in the other original equation.
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?
3-2 Reteaching (continued)
Solving Systems Algebraically
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
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.
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
3-4 Reteaching (continued)
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