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Foundations of Linear Equations – Answer Key | 1
Math 101: Foundations of Linear Equations Practice Problem Set – Answer Key
1. Which of the numbers in the following set are rational numbers? {−7, 168, 4.7, √3,2
3, π,
−2.13}
Answer: -7. 168, 4.7, 2
3, -2.13
2. Which of the numbers in the following set are irrational numbers? √2, 1.78, −9,3
4, π
Answer: √2, π
3. Which of the numbers in the following set are integers? −0.5,14, √5, 𝑖, −3,0
Answer: 14, -3, 0
4. For which line is the wrong slope given?
a) 3𝑥 + 𝑦 = 9, 𝑚 = −3
b) Two points on the line are (3, 5) and (0, 6), 𝑚 = −1
3
c) 𝑦 − 2𝑥 + 4 = 0, 𝑚 = −2
d) 𝑦 − 4 =1
2(𝑥 + 2), 𝑚 =
1
2
Answer:
3𝑥 + 𝑦 = 9 −3𝑥 − 3𝑥 𝑦 = −3𝑥 + 9
𝑚 = −3 correct
𝑚 =6 − 5
0 − 3
𝑚 =1
−3
correct
𝑦 − 2𝑥 + 4 = 0 + 2𝑥 − 4 + 2𝑥 – 4
𝑦 = 2𝑥 − 4 𝑚 = 2
incorrect
𝑦 − 4 =1
2(𝑥 + 2)
+4 + 4
𝑦 =1
2(𝑥 + 2) + 4
𝑚 =1
2
correct
5. Find the slope of the line passing through the points (-2, -3) and (-1, 5).
Answer: 𝑚 =𝑦2−𝑦1
𝑥2−𝑥1=
5−(−3)
−1−(−2)=
8
1= 8
6. What is the slope of the line 3𝑦 − 6𝑥 + 3 = 0
Answer:
3𝑦 − 6𝑥 + 3 = 0
−3 − 3
3𝑦 − 6𝑥 = −3
+6𝑥 + 6𝑥 3𝑦
3=
6𝑥 − 3
3
𝑦 = 2𝑥 − 1, 𝑚 = 2
Foundations of Linear Equations – Answer Key | 2
7. Which of the following is a solution to the function graphed below?
a) (-1, 4)
b) (-2, 0)
c) (2, 2)
d) (0, 1)
Answer: (2, 2)
Foundations of Linear Equations – Answer Key | 3
8. Which of the following is NOT a solution of the function graphed below?
a) (-4, 4)
b) (2, 0)
c) (4, 0)
d) (0, 2)
Answer: (2, 0)
Foundations of Linear Equations – Answer Key | 4
9. Which of the following is not a solution of the function 𝑓(𝑥) =1
4𝑥 + 3?
a) (8, 5)
b) (4, 4)
c) (0, 3)
d) (8, 7)
Answer: (8, 7)
𝑎) 𝑓(8) = 1
4(8) + 3
=8
4+ 3
= 2 + 3 = 5 (8, 5)
𝑏) 𝑓(4) = 1
4(4) + 3
=4
4+ 3
= 1 + 3 = 4 (4, 4)
𝑐) 𝑓(0) =1
4(0) + 3
=0
4+ 3
= 0 + 3 = 3 (0, 3)
𝑑) 𝑓(8) =1
4(8) + 3
=8
4+ 3
= 2 + 3 = 5
(8, 5) not (8, 7)
Foundations of Linear Equations – Answer Key | 5
10. What is the slope of the line below?
Answer: 3
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
=3 − 0
4 − 3
=3
1= 3
Foundations of Linear Equations – Answer Key | 6
11. What is the slope of the line below?
Answer: −1
4
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
=1 − 3
4 − 3
=−2
8=
−1
4
12. What is the slope of the line below?
Answer: 1
2
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
=0 − (−2)
4 − 0
=2
4=
1
2
Foundations of Linear Equations – Answer Key | 7
13. What is the linear equation shown below?
Answer:
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
=−3 − (−2)
0 − 1=
−5
−1= 5
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −3
y = 5x − 3
14. Which linear equation is shown below?
Answer:
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
=1 − 2
3 − 2=
−1
1= −1
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 4
y = −x + 4
Foundations of Linear Equations – Answer Key | 8
15. Find the equation of the line below.
Answer:
𝑚 =𝑦2 − 𝑦1
𝑥2 − 𝑥1
=0 − (−1)
2 − 0=
1
2
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −1
y =1
2x − 1
16. What are the x- and y-intercepts of 3𝑥 + 2𝑦 = 18?
Answer:
x-int.) 3𝑥 + 2(0) = 18
3𝑥 + 0 = 18 3𝑥
𝑥=
18
3
𝑥 = 6
y-int.) 3(0) + 2𝑦 = 18
0 + 2𝑦 = 18 2𝑦
2=
18
2
𝑦 = 9
Foundations of Linear Equations – Answer Key | 9
17. What are the x- and y-intercepts of −2𝑦 = −𝑥 + 10?
Answer:
x-int.) −2(0) = −𝑥 + 10
0 = −𝑥 + 10
−10 − 10
−10 = −𝑥
𝑥 = 10
y-int.) −2𝑦 = −(0) + 10
−2𝑦 = 0 + 10 −2𝑦
−2=
10
−2
𝑦 = −5
18. Find the x- and y-intercepts of −4𝑥 − 2𝑦 = −10.
Answer: x-int.) −4𝑥 − 2(0) = −10
−4𝑥 − 0 = −10
−4𝑥 = −10
𝑥 =−10
−4=
5
2
y-int.) −4(0) − 2𝑦 = −10
0 − 2𝑦 = −10 −2𝑦
−2=
−10
−2
𝑦 = 5
19. If the graph below is that of 𝐴𝑥 + 𝐵𝑦 = 1, what are the correct signs of A and B?
Answer: 1
𝐵 must be negative so B must be less than zero.
−𝐴
𝐵 must be negative so A must be less
than zero.
Foundations of Linear Equations – Answer Key | 10
20. If the graph of 𝐴𝑥 + 𝐵𝑦 = 1 is given below, what must the signs of A and B be?
Answer: 1
𝐵 must be positive so 𝐵 > 0.
−𝐴
𝐵 must also be positive so 𝐴 < 0.
21. If the graph of 𝐴𝑥 + 𝐵𝑦 = 1 is given below, what are the signs of A and B?
Answer: 1
𝐵 must be negative so 𝐵 < 0.
−𝐴
𝐵 must be positive so 𝐴 > 0.
Foundations of Linear Equations – Answer Key | 11
22. Graph 𝑦 = −3𝑥 + 2.
Answer:
y-int. = 2, slope = -3
23. Graph 𝑦 =1
2𝑥– 1.
Answer:
y-int. = -1, slope = 1
2
24. Graph 𝑦 = −𝑥 – 2.
Answer:
y-int. = -2, slope = -1
Foundations of Linear Equations – Answer Key | 12
25. What is the slope of the line below?
Answer: 0
26. What is the slope of the line below?
Answer: undefined
27. What is the slope of the line passing through (3, 3) and (3, -2)?
Answer: undefined 𝑚 =−2−3
3−3=
−5
0
Foundations of Linear Equations – Answer Key | 13
28. Two points on a line are (3, -2) and (4, 1).
Which statement is false?
a) The slope is m = 3.
b) The line is parallel to the line y = x – 3.
c) The line never enters the first quadrant.
d) The y-intercept of line is b = -11.
Answer: c
a) 𝑚 =𝑦2−𝑦1
𝑥2−𝑥1=
1−(−2)
4−3=
3
1= 3
b) parallel lines have the same slope
c)
d) 𝑦 = 3𝑥 + 𝑏
1 = 3(4) + 𝑏
1 = 12 + 𝑏
−12 − 12
−11 = 𝑏
29. Find the slope-intercept form of the line passing through (-2, -1) and (1, -4).
Answer: 𝑚 =−4−(−1)
1−(−2)=
−3
3= −1
𝑦 = −𝑥 + 𝑏
−4 = −1 + 𝑏
+1 + 1
−3 = 𝑏
𝑦 = −𝑥 – 3
Foundations of Linear Equations – Answer Key | 14
30. Graph the line passing through points (-1, 2) and (3, -2) to determine which quadrants it passes
through.
Answer: I, II, IV
31. A line has a slope of 𝑚 =1
4 and y-intercept of -2. Which is false?
a) The line will be parallel to 𝑦 = −4𝑥 + 2.
b) The line will be perpendicular to 𝑦 = −4𝑥 – 3.
c) The line will slope upward from left to right.
d) The rise of the line is 1.
Answer: a, parallel lines have the same slope.
32. What is the slope of a line perpendicular to 𝑦 = 3𝑥 + 2?
Answer: −1
3, slopes of perpendicular lines are opposite reciprocals.
33. What is the rise and run of the line 𝑦 = −3
4𝑥 + 2?
Answer: rise = -3 and run = 4 OR rise = 3 and rise = -4
34. Is (-3, 2) a solution of the line 𝑥 =2
3𝑦
Answer: no,
𝑥 =2
3𝑦
−3 =2
3(2)
−3 ≠4
3
Foundations of Linear Equations – Answer Key | 15
35. Is (-3,2) a solution of the line 𝑦 = 𝑥 + 5?
Answer: yes,
𝑦 = 𝑥 + 5 2 = −3 + 5
2 = 2
36. Is (-2, -1) a solution of the line 𝑦 = −2𝑥 – 5?
Answer: yes,
𝑦 = −2𝑥 − 5
−1 = −2(−2) − 5
−1 = 4 − 5
−1 = −1
37. Solve the system of equations.
Answer:
𝑥 + 3𝑦 = 5
−3𝑦 − 3𝑦
𝑥 = 5 − 3𝑦
4(5 − 3𝑦) − 2𝑦 = 6
20 − 12y − 2y = 6
−20 − 20
−14𝑦 = −14
𝑦 = 1
𝑥 + 3(1) = 5
𝑥 + 3 = 5
−3 − 3
𝑥 = 2 (2,1)
38. Find the point that is a solution to both 𝑦 − 3𝑥 = −5 and 𝑦 − 2𝑥 = −2
Answer: 𝑦 − 3𝑥 = −5
+ 3𝑥 + 3𝑥
𝑦 = 3𝑥 − 5
𝑦 − 2𝑥 = −2
3𝑥 − 5 − 2𝑥 = −2
𝑥 − 5 = −2
+5 + 5
𝑥 = 3
𝑦 − 3𝑥 = −5
𝑦 − 3(3) = −5
𝑦 − 9 = −5
+9 + 9
𝑦 = 4 (3,4)
Foundations of Linear Equations – Answer Key | 16
39. What is the solution to the system of equations?
Answer:
−2𝑥 + 2𝑦 = −8
−2𝑦 − 2𝑦 −2𝑥
−2=
−2𝑦 − 8
−2
𝑥 = 𝑦 + 4
4(𝑦 + 4) − 3𝑦 = 14
4𝑦 + 16 − 3𝑦 = 14
𝑦 + 16 = 14
−16 − 16
𝑦 = −2
−2𝑥 + 2(−2) = −8
−2𝑥 − 4 = −8
+4 + 4 −2𝑥
−2=
−4
−2
𝑥 = 2 (2, −2)
40. Zoom Car Rentals charges $25 per day, plus 35₵ per mile to rent a car. Friendly Car Rentals
charges 55₵ per mile with no upfront fee. If “c” is the total rental cost and “m” is the number of
miles, write a system of equations to represent the situation.
Answer:
𝑐 = 25 + 0.35𝑚
𝑐 = 0.55𝑚
41. The entrance fee to gain admission to Carnival A is $4 per person and the cost for a ticket to ride
an attraction is $2. Carnival B only charges $2 per person for admission, but tickets to ride cost
$2.50 each. Write a system of equations that could be used to represent the situation.
Answer:
4 + 2𝑟 = 𝑐
2 + 2.5𝑟 = 𝑐
Foundations of Linear Equations – Answer Key | 17
42. A concession stand sells hamburgers. One family orders two hamburgers and three hotdogs and
pays $20.50. The next order is for three hamburgers and two hotdogs and totals $22. Write a
system of equations and solve for the individual price of a hamburger and hotdog.
Answer:
2𝑥 + 3𝑦 = 20.5 3𝑥 + 2𝑦 = 22
3𝑥 + 2𝑦 = 22 −3𝑥 − 3𝑥 2𝑦
2=
22 − 3𝑥
2
𝑦 = 11 −3
2𝑥
2𝑥 + 3 (11 −3
2𝑥) = 20.5
2𝑥 + 33 −9
2𝑥 = 20.5
−2.5𝑥 + 33 = 20.5 −33 − 33 −2.5𝑥 = −12.5 𝑥 = 5 ℎ𝑎𝑚𝑏𝑢𝑟𝑔𝑒𝑟𝑠 = $5
3(5) + 2𝑦 = 22 15 + 2𝑦 = 22 2𝑦
2=
7
2
𝑦 = 3.5 ℎ𝑜𝑡𝑑𝑜𝑔𝑠 = $3.50
43. Solve the given system of linear equations by graphing.
Answer: (2, -1)
2𝑥 − 𝑦 = 5
−2𝑥 − 2 −𝑦
−1=
−2𝑥 + 5
−1
1
2𝑥 + 𝑦 = 0
−1
2𝑥 −
1
2𝑥
𝑦 =−1
2𝑥
𝑦 = 2𝑥 – 5
Foundations of Linear Equations – Answer Key | 18
44. Solve the system of equations using substitution.
4𝑥 + 3𝑦 = 2
−𝑥 − 2𝑦 = −3
Answer:
−𝑥 − 2𝑦 = −3
+2𝑦 + 2𝑦 −𝑥
−1=
2𝑦 − 3
−1
𝑥 = −2𝑦 + 3
4(−2𝑦 + 3) + 3𝑦 = 2
−8𝑦 + 12 + 3𝑦 = 2
−5𝑦 + 12 = 2
−12 − 12 −5𝑦
−5=
−10
−5
𝑦 = 2
−𝑥 − 2𝑦 = −3
−𝑥 − 2(2) = −3
−𝑥 − 4 = −3
−𝑥 = 1, 𝑥 = −1 (−1,2)
45. Solve the system of equation using the elimination method.
2𝑥 − 3𝑦 = −2
−4𝑥 + 𝑦 = −6
Answer:
2(2𝑥 − 3𝑦 = −2)
4𝑥 − 6𝑦 = −4
4𝑥 − 6𝑦 = −4
+(−4𝑥 + 𝑦 = -6)
−5𝑦 = −10
𝑦 = 2
2𝑥 − 3𝑦 = −2
2𝑥 − 3(2) = −2
2𝑥 − 6 = −2
+6 + 6 2𝑥
2=
4
2
(2,2)
Foundations of Linear Equations – Answer Key | 19
46. A person has 10 coins in their pocket. They are all quarters and nickels. Their total value is $1.10.
How many of each type of coin does the person have?
Answer: 𝑥 + 𝑦 = 10
. 25𝑥 + .05𝑦 = 1.10
𝑥 + 𝑦 = 10 𝑥 = 10 − 𝑦
. 25(10 − 𝑦) + 0.5𝑦 = 1.10 2.5 − .25𝑦 + .05𝑦 = 1.10 2.5 − .2𝑦 = 1.10 −.2𝑦 = −1.4 𝑦 = 7
𝑥 + 𝑦 = 10 𝑥 + 7 = 10 𝑥 = 3
3 quarters and 7 nickels.
47. A concession stand sells sodas for 75₵ and waters for $1.25. Six beverages are purchased for a
total of $5.50. How many of each type of beverage was purchased?
Answer:
𝑠 + 𝑤 = 6 → 𝑠 = 6 − 𝑤
. 75𝑠 + 1.2𝑤 = 5.50
. 75(6 − 𝑤) + 1.25𝑤 = 5.50
4.5 − .75𝑤 + 1.25𝑤 = 5.50
4.5 + .5𝑤 = 5.50
. 5𝑤 = 1
𝑤 = 2
𝑠 + 𝑤 = 6
𝑠 + 2 = 6
𝑠 = 4
2 waters and 4 sodas
48. A bakery sells cupcakes for $3.00 and cookies for $1.50. You buy 15 desserts for a party and pay
$31.50. How many of each dessert did you buy?
Answer:
𝑥 + 𝑦 = 15 → 𝑥 = 15 − 𝑦
3𝑥 + 1.5𝑦 = 31.5
3(15 − 𝑦) + 1.5𝑦 = 31.5
45 − 3𝑦 + 1.5𝑦 = 31.5
45 − 1.5𝑦 = 31.5
−1.5𝑦 = −13.5
𝑦 = 9
𝑥 + 9 = 15
𝑥 = 6
6 cupcakes and 9 cookies
