Question ID 002dba45
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 002dba45
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
ID: 002dba45 Answer
PR
Correct Answer: .1764, .1765, 3/17
Rationale
3
The correct answer is 17
. It’s given that line 𝑗 is perpendicular to line 𝑘 in the xy-plane. This means that the slope of
17
line 𝑗 is the negative reciprocal of the slope of line 𝑘. The equation of line 𝑘, 𝑦 = - + 5, is written in slope-
O
3
𝑥
intercept 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. It
17
follows that the slope of line 𝑘 is - . The negative reciprocal of a number is -1 divided by the number. Therefore, the
3
FB
17 -1 3 3
negative reciprocal of - 3 is 17 , or 17
. Thus, the slope of line 𝑗 is 17
. Note that 3/17, .1764, .1765, and 0.176 are examples
-
3
of ways to enter a correct answer.
Question Difficulty: Medium
R
O
W
N
,Question ID f224df07
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear inequalities
in one or two
variables
ID: f224df07
A cargo helicopter delivers only 100-pound packages and 120-pound packages.
For each delivery trip, the helicopter must carry at least 10 packages, and the total
weight of the packages can be at most 1,100 pounds. What is the maximum
PR
number of 120-pound packages that the helicopter can carry per trip?
A. 2
B. 4
O
C. 5
D. 6
FB
ID: f224df07 Answer
Correct Answer: C
Rationale
R
Choice C is correct. Let a equal the number of 120-pound packages, and let b equal the number of 100-pound
O
packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality
represents this situation. It’s also given that the helicopter must carry at least 10 packages:
the inequality represents this situation. Values of a and b that satisfy these two inequalities represent the
W
allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the
number of 120-pound packages, a, in the helicopter, the number of 100-pound packages, b, in the helicopter needs to
be minimized. Expressing b in terms of a in the second inequality yields , so the minimum value of b is
N
equal to . Substituting for b in the first inequality results in . Using the
distributive property to rewrite this inequality yields , or .
Subtracting 1,000 from both sides of this inequality yields . Dividing both sides of this inequality by 20
results in . This means that the maximum number of 120-pound packages that the helicopter can carry per trip
is 5.
Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.
Question Difficulty: Medium
,Question ID 3008cfc3
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 3008cfc3
PR
The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is , where and
are constants. What is the value of ?
ID: 3008cfc3 Answer
O
Correct Answer: 33
Rationale
FB
The correct answer is 33. It’s given in the table that the coordinates of two points on a line in the xy-plane are
( 𝑘, 13 ) and ( 𝑘 + 7, - 15 ) . The y-intercept is another point on the line. The slope computed using any pair of points
from the line will be the same. The slope of a line, 𝑚, between any two points, 𝑥1 , 𝑦1 and 𝑥2 , 𝑦2 , on the line can be
R
𝑦 -𝑦
calculated using the slope formula, 𝑚 = 𝑥2 - 𝑥1 . It follows that the slope of the line with the given points from the
2 1
-15 - 13 -28
table, ( 𝑘, 13 ) and ( 𝑘 + 7, - 15 ) , is 𝑚 = 𝑘+7-𝑘
, which is equivalent to 𝑚 = 7
, or 𝑚 = - 4. It's given that the y-intercept
of the line is ( 𝑘 - 5, 𝑏 ) . Substituting -4 for 𝑚 and the coordinates of the points ( 𝑘 - 5, 𝑏 ) and ( 𝑘, 13 ) into the slope
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13 - 𝑏 13 - 𝑏 13 - 𝑏
formula yields -4 = , which is equivalent to -4 = , or -4 = . Multiplying both sides of this equation by 5
𝑘-𝑘-5 𝑘-𝑘+5 5
yields -20 = 13 - 𝑏. Subtracting 13 from both sides of this equation yields -33 = - 𝑏. Dividing both sides of this
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equation by -1 yields 𝑏 = 33. Therefore, the value of 𝑏 is 33.
Question Difficulty: Hard
N
, Question ID d1b66ae6
Assessment Test Domain Skill Difficulty
SAT Math Algebra Systems of two
linear equations in
two variables
ID: d1b66ae6
If satisfies the system of equations
PR
above, what is the value of y ?
ID: d1b66ae6 Answer
O
Rationale
FB
The correct answer is . One method for solving the system of equations for y is to add corresponding sides of the
two equations. Adding the left-hand sides gives , or 4y. Adding the right-hand sides yields
. It follows that . Finally, dividing both sides of by 4 yields or . Note that 3/2
R
and 1.5 are examples of ways to enter a correct answer.
O
Question Difficulty: Hard
W
N
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 002dba45
Line is defined by . Line is perpendicular to line in the xy-plane. What is the slope of line ?
ID: 002dba45 Answer
PR
Correct Answer: .1764, .1765, 3/17
Rationale
3
The correct answer is 17
. It’s given that line 𝑗 is perpendicular to line 𝑘 in the xy-plane. This means that the slope of
17
line 𝑗 is the negative reciprocal of the slope of line 𝑘. The equation of line 𝑘, 𝑦 = - + 5, is written in slope-
O
3
𝑥
intercept 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. It
17
follows that the slope of line 𝑘 is - . The negative reciprocal of a number is -1 divided by the number. Therefore, the
3
FB
17 -1 3 3
negative reciprocal of - 3 is 17 , or 17
. Thus, the slope of line 𝑗 is 17
. Note that 3/17, .1764, .1765, and 0.176 are examples
-
3
of ways to enter a correct answer.
Question Difficulty: Medium
R
O
W
N
,Question ID f224df07
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear inequalities
in one or two
variables
ID: f224df07
A cargo helicopter delivers only 100-pound packages and 120-pound packages.
For each delivery trip, the helicopter must carry at least 10 packages, and the total
weight of the packages can be at most 1,100 pounds. What is the maximum
PR
number of 120-pound packages that the helicopter can carry per trip?
A. 2
B. 4
O
C. 5
D. 6
FB
ID: f224df07 Answer
Correct Answer: C
Rationale
R
Choice C is correct. Let a equal the number of 120-pound packages, and let b equal the number of 100-pound
O
packages. It’s given that the total weight of the packages can be at most 1,100 pounds: the inequality
represents this situation. It’s also given that the helicopter must carry at least 10 packages:
the inequality represents this situation. Values of a and b that satisfy these two inequalities represent the
W
allowable numbers of 120-pound packages and 100-pound packages the helicopter can transport. To maximize the
number of 120-pound packages, a, in the helicopter, the number of 100-pound packages, b, in the helicopter needs to
be minimized. Expressing b in terms of a in the second inequality yields , so the minimum value of b is
N
equal to . Substituting for b in the first inequality results in . Using the
distributive property to rewrite this inequality yields , or .
Subtracting 1,000 from both sides of this inequality yields . Dividing both sides of this inequality by 20
results in . This means that the maximum number of 120-pound packages that the helicopter can carry per trip
is 5.
Choices A, B, and D are incorrect and may result from incorrectly creating or solving the system of inequalities.
Question Difficulty: Medium
,Question ID 3008cfc3
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 3008cfc3
PR
The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is , where and
are constants. What is the value of ?
ID: 3008cfc3 Answer
O
Correct Answer: 33
Rationale
FB
The correct answer is 33. It’s given in the table that the coordinates of two points on a line in the xy-plane are
( 𝑘, 13 ) and ( 𝑘 + 7, - 15 ) . The y-intercept is another point on the line. The slope computed using any pair of points
from the line will be the same. The slope of a line, 𝑚, between any two points, 𝑥1 , 𝑦1 and 𝑥2 , 𝑦2 , on the line can be
R
𝑦 -𝑦
calculated using the slope formula, 𝑚 = 𝑥2 - 𝑥1 . It follows that the slope of the line with the given points from the
2 1
-15 - 13 -28
table, ( 𝑘, 13 ) and ( 𝑘 + 7, - 15 ) , is 𝑚 = 𝑘+7-𝑘
, which is equivalent to 𝑚 = 7
, or 𝑚 = - 4. It's given that the y-intercept
of the line is ( 𝑘 - 5, 𝑏 ) . Substituting -4 for 𝑚 and the coordinates of the points ( 𝑘 - 5, 𝑏 ) and ( 𝑘, 13 ) into the slope
O
13 - 𝑏 13 - 𝑏 13 - 𝑏
formula yields -4 = , which is equivalent to -4 = , or -4 = . Multiplying both sides of this equation by 5
𝑘-𝑘-5 𝑘-𝑘+5 5
yields -20 = 13 - 𝑏. Subtracting 13 from both sides of this equation yields -33 = - 𝑏. Dividing both sides of this
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equation by -1 yields 𝑏 = 33. Therefore, the value of 𝑏 is 33.
Question Difficulty: Hard
N
, Question ID d1b66ae6
Assessment Test Domain Skill Difficulty
SAT Math Algebra Systems of two
linear equations in
two variables
ID: d1b66ae6
If satisfies the system of equations
PR
above, what is the value of y ?
ID: d1b66ae6 Answer
O
Rationale
FB
The correct answer is . One method for solving the system of equations for y is to add corresponding sides of the
two equations. Adding the left-hand sides gives , or 4y. Adding the right-hand sides yields
. It follows that . Finally, dividing both sides of by 4 yields or . Note that 3/2
R
and 1.5 are examples of ways to enter a correct answer.
O
Question Difficulty: Hard
W
N
