Solving Systems Using
Tables and Graphs
3-1
Vocabulary
Review
1. Cross out the equation that is NOT in slope-intercept form.
y 5 217 x
a 5 !3b 1 5
r5s
3x 1 7y 5 13
Vocabulary Builder
linear system
linear system (noun)
LIN
ee ur SIS tum
5x 7y 1
4x y 9
Related Words: independent system, dependent system
Use Your Vocabulary
The graphs below show the possible types of solutions for a system of two equations
in two variables. Write T for true or F for false.
Intersecting Lines
y
Coinciding Lines
y
x
x
O
one solution
Consistent
Independent
Parallel Lines
y
O
O
infinitely many solutions
Consistent
Dependent
no solution
Inconsistent
F
2. Inconsistent linear systems intersect at two points.
T
3. An independent linear system has one solution.
F
4. A dependent linear system has no solutions.
T
5. Two unique lines with the same slope form an inconsistent system.
Chapter 3
58
x
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Definition: A linear system is a collection of linear equations involving the same set
of variables. The system above is two equations in two variables.
Problem 1 Using a Graph or Table to Solve a System
e
Got It? What is the solution of the system?
3x 2 2y 5 4
3x 1 2y 5 5
6. Circle the graph of the equations.
y
y
y
4
4
2
2
x
4 2
O
2
2
x
4
4 2
2
O
2
x
4 2
4
2
O
2
4
2
4
7. Circle the row in the calculator screen
that contains the solution of the system.
X
Y1
Y2
0
1
2
3
4
–2
–1.5
–1
–.5
0
5
2
–1
–4
–7
8. The solution of the system of equations is Q 2 , 21 R .
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Problem 2 Using a Table to Solve a Problem
Got It? Biology The equation y1 5 1.5x 1 22 models the length, in centimeters, of
a Spiny Dogfish shark x years old. The equation y2 5 0.75x 1 37 models the length,
in centimeters, of a Greenland shark x years old. If the growth rates continue, how
long will each shark be when it is 25 years old?
9. Reasoning Which shark will be longer at age 25? How do you know?
Answers may vary. Sample: The Spiny Dogfish has a faster growth
_______________________________________________________________________
rate, so it will be longer at age 25.
_______________________________________________________________________
10. Underline the correct phrase to complete the sentence.
To solve the problem, I need to find
an x-value / a y-value / two x-values / two y-values .
11. Complete the table for x 5 25.
x
y1 1.5x 22
y1 0.75x 37
25
59.5
55.75
59
Lesson 3-1
12. Spiny Dogfish sharks will be
59.5
cm long and Greenland sharks will be
55.75 cm long when they are 25 years old.
Problem 3
Using Linear Regression
Got It? The table shows the populations of the San Diego and Detroit metropolitan
regions. When were the populations of these regions equal? What was that population?
Populations of San Diego and Detroit (1950–2000)
San Diego
Detroit
1950
1960
1970
1980
1990
2000
334,387
573,224
696,769
875,538
1,110,549
1,223,400
1,849,568
1,670,144
1,511,482
1,203,339
1,027,974
951,270
SOURCE: U.S. Census Bureau
13. Circle the first calculator step in solving the problem.
Calculate the intersection.
Enter the data into lists.
Enter y1 .
14. Write the name of the list you will use (L1 , L2 , or L3 ) next to the data type.
L2
population of San Diego
L3
population of Detroit
L1
years since 1950
y1 5 17816.597 cx 1 356896.238 c
y1 5 217816.597x c 1 356896.238 c
y2 5 219217.551 cx 1 1849401.619 c
y2 5 19217.551x c 1 1849401.619 c
16. Cross out the graph that does NOT show the regression lines.
17. Underline the correct word to complete each sentence.
The x-axis / y-axis corresponds to the number of years since 1950.
The x-axis / y-axis corresponds to the populations of San Diego and Detroit.
18. The scale of the graphs is 260 # x # 60 by 10s and 0 # y # 1,500,000 by 100,000s.
Use the scale to estimate the coordinates of the point of intersection. Accept reasonable estimates.
Exact answers are given.
1,074,919
The coordinates are Q 40.3 ,
R.
The populations of San Diego and Detroit were equal sometime during the year 1990 .
The population was about
Chapter 3
1,075,000
.
60
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
15. Circle the pair of equations you will graph.
Problem 4 Classifying a System Without Graphing
23x 1 y 5 4
Got It? Without graphing, is the system e x 2 1 y 5 1
3
independent, dependent,
or inconsistent?
19. Write each equation in slope-intercept form.
x 2 13 y 5 1
y 5 3x 1 4
23x 1 y 5 4
y 5 3x 2 3
20. The slope of 23x 1 y 5 4 is 3 and the slope of x 2 31 y 5 1 is 3 .
21. The y-intercept of 23x 1 y 5 4 is 4 and the y-intercept of x 2 13 y 5 1 is 23 .
22. Underline the correct words to complete the sentence.
Because the slopes of the lines are equal / not equal and the y-intercepts are
the same / different , the system is inconsistent / independent / dependent .
Lesson Check • Do you UNDERSTAND?
Vocabulary Is it possible for a system of equations to be both independent and
inconsistent? Explain.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Write T for true or F for false.
T
23. The graphs of an inconsistent system are parallel.
T
24. The graphs of an independent system intersect at one point.
25. Now answer the question.
Answers may vary. Sample: No; inconsistent lines are parallel and
_______________________________________________________________________
never intersect. Independent lines have one point of intersection.
_______________________________________________________________________
Therefore, a system cannot be both independent and inconsistent.
_______________________________________________________________________
Math Success
Check off the vocabulary words that you understand.
system of equations
dependent
independent
consistent
inconsistent
Rate how well you can solve a linear system using a graph or table.
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61
Lesson 3-1