Page 1
Question ID: 3cdbf026
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
The graph of the equation is a line in the xy-plane, where a and k are constants. If the line contains the points and
, what is the value of k ?
Answer
A.
B.
C. 2
D. 3
Correct Answer: A
Rationale
Choice A is correct. The value of k can be found using the slope-intercept form of a linear equation, , where m is the slope and b is
the y-coordinate of the y-intercept. The equation can be rewritten in the form . One of the given points,
, is the y-intercept. Thus, the y-coordinate of the y-intercept must be equal to . Multiplying both sides by k gives .
Dividing both sides by gives .
Choices B, C, and D are incorrect and may result from errors made rewriting the given equation.
Page 2
Question ID: 9bbce683
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
For line , the table shows three values of and their corresponding values of . Line is the result of translating line down units in the xy-
plane. What is the x-intercept of line ?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. The equation of line can be written in slope-intercept form , where is the slope of the line and is the
y
-intercept of the line. It’s given that line contains the points , , and . Therefore, its slope can be found as
, or . Substituting for in the equation yields . Substituting for and for in this equation yields
, or . Subtracting from both sides of this equation yields . Substituting for in
yields . Since line is the result of translating line down units, an equation of line is , or .
Substituting for in this equation yields . Solving this equation for yields . Therefore, the x-intercept of line is
îš•
.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Page 3
Question ID: 7625073d
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Medium
Question
The equation represents the number of blue tiles, , and the number of green tiles, , an artist needs for an -square-inch tile
project. The artist needs blue tiles for the project. How many green tiles does he need?
Correct Answer: 49
Rationale
The correct answer is . It’s given that the equation represents the number of blue tiles, , and the number of green tiles, , an
artist needs for an -square-inch tile project. It’s also given that the artist needs blue tiles for the project. Substituting for in the
equation yields , or . Subtracting from both sides of this equation yields .
Dividing both sides of this equation by yields . Therefore, the artist needs green tiles for the project.
Page 4
Question ID: fdee0fbf
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
In the xy-plane, line k intersects the y-axis at the point and passes through the point . If the point lies on line k, what is
the value of w ?
Rationale
The correct answer is 74. The y-intercept of a line in the xy-plane is the ordered pair of the point of intersection of the line with the y-axis.
Since line k intersects the y-axis at the point , it follows that is the y-intercept of this line. An equation of any line in the xy-plane
can be written in the form , where m is the slope of the line and b is the y-coordinate of the y-intercept. Therefore, the equation of
line k can be written as , or . The value of m can be found by substituting the x- and y-coordinates from a point on
the line, such as , for x and y, respectively. This results in . Solving this equation for m gives . Therefore, an equation of
line k is . The value of w can be found by substituting the x-coordinate, 20, for x in the equation of line k and solving this equation for
y. This gives , or . Since w is the y-coordinate of this point, .
Page 5
Question ID: 9aaf7786
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
In the xy-plane, line has a slope of and an x-intercept of . What is the y-coordinate of the y-intercept of line ?
Correct Answer: -10
Rationale
The correct answer is . A line in the xy-plane can be represented by the equation , where is the slope of the line and is the y-
coordinate of the y-intercept. It's given that line has a slope of . Therefore, . It's also given that line has an x-intercept of
. Therefore, when , . Substituting for , for , and for in the equation yields
, which is equivalent to
îš•
. Subtracting from both sides of this equation yields . Therefore, the y-coordinate of the y-
intercept of line is .
Page 6
Question ID: 0b46bad5
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
In the equation above, a and b are constants and . Which of the following could represent the graph of the equation in the xy-plane?
Answer
A.
B.
C.
D.
Correct Answer: C
Page 7
Rationale
Choice C is correct. The given equation can be rewritten in slope-intercept form, , where m represents the slope of
the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the
equation yields , and dividing both sides of this equation by b yields , or . With the
equation now in slope-intercept form, it shows that , which means the y-coordinate of the y-intercept is 1. It’s given that a and b are both
greater than 0 (positive) and that . Since , the slope of the line must be a value between and 0. Choice C is the only graph
of a line that has a y-value of the y-intercept that is 1 and a slope that is between and 0.
Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between and 0.
Page 8
Question ID: 94b48cbf
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
The graph of in the -plane has an -intercept at and a -intercept at , where and are constants. What is the
value of ?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. The x-coordinate of the x-intercept can be found by substituting for in the given equation, which gives
îš•
, or . Dividing both sides of this equation by yields . Therefore, the value of is . The y-coordinate of the y-
intercept can be found by substituting for in the given equation, which gives , or . Dividing both sides of
this equation by
yields . Therefore, the value of is
îš•
. It follows that the value of is , which is equivalent to
îš•
, or
.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Page 9
Question ID: df78b361
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Medium
Question
Lily made cups of jam. Lily then filled small containers and large containers with all the jam she made. The equation
represents this situation. Which is the best interpretation of in this context?
Answer
A. The number of large containers Lily filled
B. The number of small containers Lily filled
C. The total number of cups of jam in the large containers
D. The total number of cups of jam in the small containers
Correct Answer: C
Rationale
Choice C is correct. It’s given that the equation represents the situation where Lily filled small containers and large containers
with all the jam she made, which was cups. Therefore, represents the total number of cups of jam in the large containers.
Choice A is incorrect. The number of large containers Lily filled is represented byîš•, notîš•.
Choice B is incorrect. The number of small containers Lily filled is represented by , notîš•.
Choice D is incorrect. The total number of cups of jam in the small containers is represented byîš•, notîš•.
Page 10
Question ID: 98d3393a
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
Line in the xy-plane is perpendicular to the line with equation . What is the slope of line ?
Answer
A. 0
B.
C.
D. The slope of line is undefined.
Correct Answer: A
Rationale
Choice A is correct. It is given that line is perpendicular to a line whose equation is x = 2. A line whose equation is a constant value of x is
vertical, so must therefore be horizontal. Horizontal lines have a slope of 0, so has a slope of 0.
Choice B is incorrect. A line with slope
îš•
is perpendicular to a line with slope 2. However, the line with equation x = 2 is vertical and has
undefined slope (not slope of 2).Choice C is incorrect. A line with slope –2 is perpendicular to a line with slope . However, the line with
equation x = 2 has undefined slope (not slope of ). Choice D is incorrect; this is the slope of the line x = 2 itself, not the slope of a line
perpendicular to it.
Page 11
Question ID: cc3e9528
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
The graph of is translated down units in the xy-plane. What is the x-coordinate of the x-intercept of the resulting graph?
Correct Answer: 59/9, 6.555, 6.556
Rationale
The correct answer is . When the graph of an equation in the form , where , , and are constants, is translated down
units in theîš•xy
-plane, the resulting graph can be represented by the equation
îš•
. It’s given that the graph of
is translated down units in the xy-plane. Therefore, the resulting graph can be represented by the equation , or
îš•
. Adding to both sides of this equation yields . The x-coordinate of the x-intercept of the graph of an equation in
the xy-plane is the value of in the equation when . Substituting for in the equation yields , or
. Dividing both sides of this equation by yields . Therefore, the x-coordinate of the x-intercept of the resulting graph is .
Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.
Page 12
Question ID: c4ea43ef
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B.
In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of
numbers of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?
Answer
A. 128
B. 160
C. 200
D. 320
Correct Answer: B
Rationale
Choice B is correct. Avery earns $10 per hour working at job A. Therefore, if she works a hours at job A, she will earn dollars. Avery earns
$20 per hour working at job B. Therefore, if she works b hours at job B, she will earn dollars. The graph shown represents all possible
combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. Therefore, if she worked a hours at job A, worked
b hours at job B, and earned s dollars from both jobs, the following equation represents the graph: , where s is a constant.
Identifying any point from the graph and substituting the values of the coordinates for a and b, respectively, in this equation yield the value
of s. For example, the point , where and , lies on the graph. Substituting 16 for a and 0 for b in the equation
yields , or . Similarly, the point , where and , lies on the graph.
Substituting 0 for a and 8 for b in the equation yields , or .
Choices A, C, and D are incorrect. If the value of s is 128, 200, or 320, then no points on the graph will satisfy this equation. For example, if
the value of s is 128 (choice A), then the equation becomes . The point , where and
, lies on the graph. However, substituting 16 for a and 0 for b in yields , or ,
which is false. Therefore, doesn’t satisfy the equation, and so the value of s can’t be 128. Similarly, if (choice C) or
(choice D), then substituting 16 for a and 0 for b yields and , respectively, which are both false.
Page 13
Question ID: a1fd2304
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?
Rationale
The correct answer is 1.5. The total amount, in liters, of a saline solution can be expressed as the liters of each type of saline solution multiplied
by the percent concentration of the saline solution. This gives , , and , where x is the amount, in liters, of 25%
saline solution and 10%, 15%, and 25% are represented as 0.10, 0.15, and 0.25, respectively. Thus, the equation
must be true. Multiplying 3 by 0.10 and distributing 0.15 to yields
. Subtracting 0.15x and 0.30 from each side of the equation gives . Dividing each side of the
equation by 0.10 yields . Note that 1.5 and 3/2 are examples of ways to enter a correct answer.
Page 14
Question ID: 8a1544f1
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Easy
Question
What is the equation of the line shown in the xy-plane above?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. Any line in the xy-plane can be defined by an equation in the form y = mx + b, where m is the slope of the line and b is the y-
coordinate of the y-intercept of the line. From the graph, the y-intercept of the line is (0, 3). Therefore, b = 3. The slope of the line is the change in
the value of y divided by the change in the value of x for any two points on the line. The line shown passes through (0, 3) and (1, 0), so
, or m = –3. Therefore, the equation of the line is y = –3x + 3.
Choices A and C are incorrect because the equations given in these choices represent a line with a positive slope. However, the line shown has a
negative slope. Choice D is incorrect because the equation given in this choice represents a line with slope of . However, the line shown
has a slope of –3.
Page 15
Question ID: 00b9bd37
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Medium
Question
20 40 60 80 100
x
10
20
30
40
50
y
O
Number of T-shirts
Number of sweatshirts
The graph models the relationship between the number of T-shirts, , and the number of sweatshirts, , that Kira can purchase for a school
fundraiser. Which equation could represent this relationship?
Answer
A.
B.
C.
D.
Correct Answer: B
Rationale
Choice B is correct. A line in the xy-plane can be written as , where is the slope of the line and is the y-coordinate of the y-
intercept. The graph shown is a line passing through the points and . Substituting for and for in the equation
yields , or . Substituting for , for , and for in the equation yields .
Subtracting from both sides of this equation yields . Dividing both sides of this equation by yields , or .
Substituting for and for in the equation yields . Multiplying both sides of this equation by yields
, or . Adding to both sides of this equation yields . Therefore, the equation
represents the relationship between and modeled by the graph.
Choice A is incorrect. The point is not on the graph of this equation, since , not .
Choice C is incorrect. The point is not on the graph of this equation, since , not .
Choice D is incorrect. The point is not on the graph of this equation, since , not .
Page 16
Question ID: 49800634
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in two
variables
Hard
Question
The table shows two values of and their corresponding values of . In the xy-plane, the graph of the linear equation representing this
relationship passes through the point . What is the value of ?
Answer
A.
B.
C.
D.
Correct Answer: D
Rationale
Choice D is correct. The linear relationship between and can be represented by the equation , where is the slope of the graph
of this equation in the xy-plane and is the y-coordinate of the y-intercept. The slope of a line between any two points and on
the line can be calculated using the slope formula . Based on the table, the graph contains the points and .
Substituting
and for and , respectively, in the slope formula yields , which is equivalent to
, or . Substituting for , for , and for in the equation yields , or
. Adding to both sides of this equation yields . Therefore, and . Substituting for and for in the equation
yields . Thus, the equation represents the linear relationship between and . It's also given that the
graph of the linear equation representing this relationship in the xy-plane passes through the point . Substituting for and for in the
equation yields , which is equivalent to , or .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
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