Question ID 3008cfc3
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 3008cfc3
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
Correct Answer: 33
Rationale
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 calculated using the
(𝑦 − 𝑦1)
slope formula, 𝑚 = 2 . It follows that the slope of the line with the given points from the table, (𝑘, 13) and
(𝑥−15
2 − 𝑥−
1)13 −28
(𝑘 + 7, − 15), is 𝑚 = , which is equivalent to 𝑚 = , or 𝑚 = − 4. It's given that the y-intercept of the line is
𝑘+7−𝑘 7
(𝑘 − 5, 𝑏). Substituting −4 for 𝑚 and the coordinates of the points (𝑘 − 5, 𝑏) and (𝑘, 13) into the slope formula yields
13 − 𝑏 13 − 𝑏 13 − 𝑏
−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
equation by −1 yields 𝑏 = 33. Therefore, the value of 𝑏 is 33.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 1 из 119
, 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
above, what is the value of y ?
ID: d1b66ae6 Answer
Rationale
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
and 1.5 are examples of ways to enter a correct answer.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 2 из 119
, Question ID 3cdbf026
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 3cdbf026
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 ?
A.
B.
C. 2
D. 3
ID: 3cdbf026 Answer
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.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 3 из 119
, Question ID ff501705
Assessment Test Domain Skill Difficulty
SAT Math Algebra Systems of two
linear equations in
two variables
ID: ff501705
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
ID: ff501705 Answer
Correct Answer: 6
Rationale
The correct answer is 6. A system of two linear equations in two variables, 𝑥 and 𝑦, has no solution if the lines
represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form,
𝐴𝑥 + 𝐵𝑦 = 𝐶 and 𝐷𝑥 + 𝐸𝑦 = 𝐹, are parallel if the coefficients for 𝑥 and 𝑦 in one equation are proportional to the
corresponding coefficients in the other equation, meaning 𝐷 = 𝐸 ; and the lines are distinct if the constants are not
𝐹 𝐷 𝐸 𝐴 𝐵 3 1 2 3
proportional, meaning is not equal to 𝑦 − 𝑥 = − 𝑦. Multiplying
or . The first equation in the given system is
𝐶 𝐴 𝐵 2 4 3 2
each side of this equation by 12 yields 18𝑦 − 3𝑥 = 8 − 18𝑦. Adding 18𝑦 to each side of this equation yields 36𝑦 − 3𝑥 = 8,
1 3 9
or −3𝑥 + 36𝑦 = 8. The second equation in the given system is 𝑥 + = 𝑝𝑦 + . Multiplying each side of this equation by
2 2 2
2 yields 𝑥 + 3 = 2𝑝𝑦 + 9. Subtracting 2𝑝𝑦 from each side of this equation yields 𝑥 + 3 − 2𝑝𝑦 = 9. Subtracting 3 from
each side of this equation yields 𝑥 − 2𝑝𝑦 = 6. Therefore, the two equations in the given system, written in standard
form, are −3𝑥 + 36𝑦 = 8 and 𝑥 − 2𝑝𝑦 = 6. As previously stated, if this system1has no solution,
−2𝑝 1 the lines
𝑝 represented by
the equations in the xy-plane are parallel and distinct, meaning the proportion = , or − = − , is true and the
6 1 6 1 −3 36 3 18 1 𝑝
proportion = is not true. The proportion = is not true. Multiplying each side of the true proportion, − = −
8 −3 8 −3 3 18
, by −18 yields 6 = 𝑝. Therefore, if the system has no solution, then the value of 𝑝 is 6.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 4 из 119
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 3008cfc3
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
Correct Answer: 33
Rationale
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 calculated using the
(𝑦 − 𝑦1)
slope formula, 𝑚 = 2 . It follows that the slope of the line with the given points from the table, (𝑘, 13) and
(𝑥−15
2 − 𝑥−
1)13 −28
(𝑘 + 7, − 15), is 𝑚 = , which is equivalent to 𝑚 = , or 𝑚 = − 4. It's given that the y-intercept of the line is
𝑘+7−𝑘 7
(𝑘 − 5, 𝑏). Substituting −4 for 𝑚 and the coordinates of the points (𝑘 − 5, 𝑏) and (𝑘, 13) into the slope formula yields
13 − 𝑏 13 − 𝑏 13 − 𝑏
−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
equation by −1 yields 𝑏 = 33. Therefore, the value of 𝑏 is 33.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 1 из 119
, 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
above, what is the value of y ?
ID: d1b66ae6 Answer
Rationale
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
and 1.5 are examples of ways to enter a correct answer.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 2 из 119
, Question ID 3cdbf026
Assessment Test Domain Skill Difficulty
SAT Math Algebra Linear equations in
two variables
ID: 3cdbf026
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 ?
A.
B.
C. 2
D. 3
ID: 3cdbf026 Answer
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.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 3 из 119
, Question ID ff501705
Assessment Test Domain Skill Difficulty
SAT Math Algebra Systems of two
linear equations in
two variables
ID: ff501705
In the given system of equations, is a constant. If the system has no solution, what is the value of ?
ID: ff501705 Answer
Correct Answer: 6
Rationale
The correct answer is 6. A system of two linear equations in two variables, 𝑥 and 𝑦, has no solution if the lines
represented by the equations in the xy-plane are parallel and distinct. Lines represented by equations in standard form,
𝐴𝑥 + 𝐵𝑦 = 𝐶 and 𝐷𝑥 + 𝐸𝑦 = 𝐹, are parallel if the coefficients for 𝑥 and 𝑦 in one equation are proportional to the
corresponding coefficients in the other equation, meaning 𝐷 = 𝐸 ; and the lines are distinct if the constants are not
𝐹 𝐷 𝐸 𝐴 𝐵 3 1 2 3
proportional, meaning is not equal to 𝑦 − 𝑥 = − 𝑦. Multiplying
or . The first equation in the given system is
𝐶 𝐴 𝐵 2 4 3 2
each side of this equation by 12 yields 18𝑦 − 3𝑥 = 8 − 18𝑦. Adding 18𝑦 to each side of this equation yields 36𝑦 − 3𝑥 = 8,
1 3 9
or −3𝑥 + 36𝑦 = 8. The second equation in the given system is 𝑥 + = 𝑝𝑦 + . Multiplying each side of this equation by
2 2 2
2 yields 𝑥 + 3 = 2𝑝𝑦 + 9. Subtracting 2𝑝𝑦 from each side of this equation yields 𝑥 + 3 − 2𝑝𝑦 = 9. Subtracting 3 from
each side of this equation yields 𝑥 − 2𝑝𝑦 = 6. Therefore, the two equations in the given system, written in standard
form, are −3𝑥 + 36𝑦 = 8 and 𝑥 − 2𝑝𝑦 = 6. As previously stated, if this system1has no solution,
−2𝑝 1 the lines
𝑝 represented by
the equations in the xy-plane are parallel and distinct, meaning the proportion = , or − = − , is true and the
6 1 6 1 −3 36 3 18 1 𝑝
proportion = is not true. The proportion = is not true. Multiplying each side of the true proportion, − = −
8 −3 8 −3 3 18
, by −18 yields 6 = 𝑝. Therefore, if the system has no solution, then the value of 𝑝 is 6.
Question Difficulty: Hard
https://satsuitequestionbank.collegeboard.org/digital/results 03/03/24, 17:47
Стр. 4 из 119
