3.1 Reading Graphs: Linear Equations in Two Variables
1. Plot the following ordered pairs: A = (3, −2), B = (5, 0), C = (0, 4)
4
C
2
B
-6
-4
2
-2
-2
4
6
A
-4
2. Determine whether the given point is a solution to the given equation
(a) 5x − 3y = 15; (5, 2)
Answer: No, it is not.
(b) y = −x/4; (−8, 2)
Answer: Yes, it is.
3. If (6, Q) is a solution to the equation 3x + 4y = 24, what is the value
of Q?
3
Answer: Q = = 1.5
2
4. If (a, −7) is a solution to the equation −5x + y = 10, what is the value
of a?
17
Answer: a = − = −3.4
5
1
5. Fill in the table below of solutions to the equation 4x − 9y = −36
x
−9
0
9
2
y
0
4
8
6. Use the graph below to answer the following questions
Population in Millions
U.S. Population, 1790-2000
250
200
150
100
50
0
1800
1850
1900
Year
1950
(a) Approximately what was the U.S. population in 1850. Express
this as an ordered pair.
Answer: About 30 million people.
(b) Approximately when was the U.S. population 100 million. Express
this as an ordered pair.
Answer: In roughly 1920.
3.2
Graphing Linear Equations in Two Variables
7. Find three ordered pairs that satisfy the equation −4x + 8y = 32.
Answer: (0, 4); (−8, 0); (2, 5). Many other answers are possible.
3
2000
8. What is the x−intercept of the line 2x − 4y = 30?
Answer: (15, 0)
9. What is the y−intercept of the line 5x − 3y = −27?
Answer: (0, 9)
10. Graph each of the following lines
(a) x − y = 6
4
2
-4
2
-2
4
6
8
10
-2
-4
-6
-8
-10
(b) 3x + 7y = 14
4
2
-4
2
-2
-2
-4
(c) x = 3
4
4
4
2
-4
2
-2
4
-2
-4
(d) y = 4
4
2
-4
-2
2
4
2
4
-2
-4
(e) 6x − 3y = 0
4
2
-4
-2
-2
-4
5
3.3
Slope of a Line
11. Find the slope of the line through the marked points. Also show the
rise and run on the plot.
4
2
A
-6
-4
2
-2
B
4
6
-2
-4
1
2
12. Find the slope of the line through the marked points. Also show the
rise and run on the plot.
Answer:
4
A
-6
-4
2
2
-2
-2
-4
6
4
6
B
Answer: −
4
9
13. Find the slope of the line through each given pair of points
(a) (9, 5) and (−14, −3).
−3 − 5
8
Answer:
=−
−14 − 9
25
(b) (−4, 6) and (7, 6).
Answer: 0
(c) (−3, 2) and (−3, 10)
Answer: The slope is undefined.
14. What is the slope of the line 6x + 9y = 54?
2
Answer: −
3
15. Find the slope of a line that is parallel to 5x − 3y = −2.
5
Answer:
3
16. Find the slope of a line that is perpendicular to 8x + 6y = −5.
3
Answer:
4
17. On the axes below, draw a line that has slope 2.
4
2
-4
2
-2
-2
-4
7
4
3.4
Equations of Lines
18. Identify the slope and y−intercept of each of the following lines
(a) 5x − y = 100
Answer: The slope is 5; the y−intercept is −100.
(b) x + 2(y − 3) = 14 − 2x
Answer: The slope is −3/2; the y−intercept is 20.
19. For each of the following, graph and find the equation of the line with
the given characteristics.
(a) slope -4, y−intercept 2.5
Answer: y = −4x + 2.5
(b) slope 1/3, y−intercept 10
x
Answer: y = − 10
3
20. For each of the following, graph and find the equation of the line with
the given characteristics.
(a) slope -4/5, passes through (3, 7)
4
23
Answer: y = − x −
5
5
(b) slope 31, passes through (−10, 1)
Answer: y = 31x + 309
21. For each of the following, graph and find the equation of the line with
the given characteristics.
(a) passes through (−2, 9) and (0, 5)
Answer: y = −2x + 5
(b) passes through (8, 0) and (4/3, 5)
3
Answer: y = − x + 6
4
(c) passes through (−2, 1) and (5, 6)
17
5
Answer: y = x +
7
7
8
3.4.1
Interesting Questions
22. Given a point in time in the past 200 years, let t be the number of
years since 1800 and P be the U.S. population. Given the data below,
is it possible that P and t are (exactly) related by a linear equation?
Years since 1800
50
100
150
Population (in millions)
32.1
70.9
151
23. A baseball factory has two types of costs–fixed costs, like rent, electricity, etc. do not depend on how many baseballs the factory produces.
Variable costs–like the material for the baseballs and labor, do depend
on how many baseballs the factory produces. Explain how one can
think of these costs as a slope and y−intercept.
3.5
Graphing Linear Inequalities in Two Variables
24. For each of the following points, indicate whether or not they satisfy
the inequality 5x + 8y ≤ 50
(a) (3, 4) does satisfy the inequality
(b) (12, 9) does not satisfy the inequality
(c) (0, 0) does satisfy the inequality
25. Graph each of the following inequalities
9
HaL 2 x - y ³ 6
20
10
-20
10
-10
20
-10
-20
HbL 3 x + 4 y < 12
10
5
-10
-5
5
10
5
10
-5
-10
HbL 5 x + 15 y > 90
10
5
-10
-5
-5
-10
10
HdL 3 x + 8 y £ 200
40
20
-40
20
-20
40
-20
-40
26. Graph each of the following inequalities
(a) y > 4
HbL y > 4
10
5
-10
5
-5
-5
-10
(b) y ≤ 2
11
10
HbL y £ 2
10
5
-10
5
-5
10
-5
-10
(c) x ≤ −1
(d) x > −2
3.5.1
More Interesting Questions
27. In the problems above, we figured out how to graph the set of all pairs
(x, y) that satisfy an inequality like y > 2. Can you graph the set of
all pairs (x, y) that satisfy the inequality y > 2 and x < 3? Can you
graph the set of all pairs (x, y) that satisfy the inequalities x ≥ 0 and
y ≥ 0.
28. Can you graph the set of pairs (x, y) that satisfy 4x + 2y ≤ 8 and x ≥ 0
and y ≥ 0?
29. Can you give a list of inequalities that describes the set of pairs shaded
below
12
6
4
2
-4
2
-2
-2
-4
-6
13
4