The SAT (Scholastic Aptitude Test) is a standardized test administered by the College Board, commonly used for college admissions in the United States. Launched in 1926, the SAT tests students' skills in writing, critical reading, and mathematics. It's offered seven times per year.

What is the cost of SAT tutoring at Northside?

Private SAT tutoring cost varies based on needs and session count. We offer flexible packages for in-person and online sessions. Visit our FAQ for bundle and pricing info.

What is the SAT score range?

SAT scores range from 400 to 1600. The test has two main sections: Math and Evidence-Based Reading and Writing. Students can score between 200 and 800 in each section.

What is the average SAT score?

The average SAT score for test-takers is around 1050 — the 50th percentile.

What scores do most students achieve?

The middle 50% of students score between 870 and 1150 (the 24th to 74th percentiles).

What is the average SAT score in Georgia?

The average SAT score in Georgia is 1067 — 533 in Math and 535 in Reading and Writing.

How can SAT tutoring improve my score?

We use practice tests to simulate real SAT conditions, identify weak areas, and develop targeted strategies. Our tutors don't just teach content — they teach proven test-taking strategies.

How long are SAT scores valid?

SAT scores are valid for five years from the test date.

When is the best time to start preparing?

Most students benefit from starting at least six months before their desired test date. Some aim for a top score by starting earlier; others achieve their goals with an intensive three-month plan.

Can I choose between online and in-person tutoring?

Yes. We offer both online and in-person tutoring with equal quality. Our tutors are experienced in delivering results through both formats.

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Scholastic Aptitude Test

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The SAT is one of the most critical elements of the college admissions process. Alongside a high school transcript, your SAT score signals to colleges how prepared you are for higher education. At Northside, we help students de-mystify the test and perform at their best on test day — whether they need a complete content review or targeted work on specific sections.

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Our SAT cohort sessions are small-group test prep classes that run through both Digital SAT sections — Math and Reading & Writing — with a professional tutor guiding strategy, timing, and content. Students benefit from peer learning while receiving direct feedback from the instructor.

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Max Pincus

Stanford University

Stanford grad, Mathematical and Computational Sciences with Economics minor. 1560 SAT, 800 Math 2. 5's on Calc AB/BC, Stats, Bio, Physics.

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Angelina Guan

Georgia Institute of Technology

Freshman at Georgia Tech majoring in Industrial Engineering. National Merit Finalist, AP Scholar with Distinction, and Westminster top 5%. Earned a 1540 SAT and 35 ACT. Received 5's on AP Calc BC, AP Stats, AP CSP, AP CSA, AP Chem, and AP Bio. Served as President of Mu Alpha Theta and received the Presidential Volunteer Service Award.

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Siya Kalra

The University of Chicago

First-year at UChicago, Business Economics and Computer Science. AP experience in Biology, Chemistry, Stats, Calc AB/BC, English, Latin.

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Georgia Institute of Technology

Westminster 2022 grad, Georgia Tech BSMS in Computer Science. 36 ACT, 1580 SAT. National Merit Scholar.

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35 Hard SAT Math Questions

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Compiled by our tutors from real Algebra and Advanced Math questions at the hardest difficulty level tested on the SAT. Attempt each one, then reveal the answer and a full explanation.

Algebra — 18 Questions

Algebra · Linear equations in one variable · Hard

Question 1. 12x + 284 − s13 = r(x − 8)
In the given equation, s and r are constants, and s > 0. If the equation has infinitely many solutions, what is the value of s?

Show Answer & Explanation

Correct Answer: 403

For a linear equation in one variable to have infinitely many solutions, the coefficients of x must be equal on both sides, and the constant terms must also be equal. Rewriting the left side gives 3x + 7 − s13 = rx − 8r. Matching x-coefficients gives r = 3. Matching constants gives 7 − s13 = −8(3) = −24, so s13 = 31, and s = 403.

Algebra · Linear equations in two variables · Hard

Question 2. The table gives the coordinates of two points on a line in the xy-plane: (k, 13) and (k + 7, −15). The y-intercept of the line is (k − 5, b), where k and b are constants. What is the value of b?

Show Answer & Explanation

Correct Answer: 33

The slope between the two given points is −15 − 13(k + 7) − k = −287 = −4. Using the point (k, 13) and the y-intercept (k − 5, b): −4 = 13 − b5. Multiplying both sides by 5 gives −20 = 13 − b, so b = 33.

Algebra · Linear equations in two variables · Hard

Question 3. The graph of the equation ax + ky = 6 is a line in the xy-plane, where a and k are constants. If the line contains the points (−2, −6) and (0, −3), what is the value of k?

A. −2
B. −1
C. 2
D. 3

Show Answer & Explanation

Correct Answer: A

Rewriting ax + ky = 6 in slope-intercept form gives y = −akx + 6k. The point (0, −3) is the y-intercept, so 6k = −3. Multiplying both sides by k gives 6 = −3k, so k = −2.

Algebra · Linear equations in one variable · Hard

Question 4. Hector used a tool called an auger to remove corn from a storage bin at a constant rate. The bin contained 24,000 bushels of corn when Hector began to use the auger. After 5 hours of using the auger, 19,350 bushels of corn remained in the bin. If the auger continues to remove corn at this rate, what is the total number of hours Hector will have been using the auger when 12,840 bushels of corn remain in the bin?

A. 3
B. 7
C. 8
D. 12

Show Answer & Explanation

Correct Answer: D

In 5 hours the auger removed 24,000 − 19,350 = 4,650 bushels, a rate of 4,6505 = 930 bushels per hour. Solving 24,000 − 930x = 12,840 gives x = 12.

Algebra · Systems of two linear equations in two variables · Hard

Question 5. ax + by = 72
6x + 2by = 56
In the given system of equations, a and b are constants. The graphs of these equations in the xy-plane intersect at the point (4, y). What is the value of a?

A. 3
B. 4
C. 6
D. 14

Show Answer & Explanation

Correct Answer: D

Multiplying the first equation by −2 gives −2ax − 2by = −144. Adding this to the second equation gives (−2a + 6)x = −88. Since x = 4 at the intersection, (−2a + 6)(4) = −88, so −2a + 6 = −22, giving a = 14.

Algebra · Linear functions · Hard

Question 6. The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Which of the following equations gives the cost y, in dollars, of renting the backhoe for x days, where x is a positive integer and x ≤ 10?

A. y = 270x − 135
B. y = 270x + 135
C. y = 135x + 270
D. y = 135x + 135

Show Answer & Explanation

Correct Answer: D

The cost is $270 for the first day plus $135 for each of the remaining (x − 1) days: y = 270 + 135(x − 1), which simplifies to y = 135x + 135.

Algebra · Linear inequalities in one or two variables · Hard

Question 7. Adam's school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, w, that Adam waits for the bus varies between 0 and 30. Which of the following inequalities gives the values of w for which it would be faster for Adam to walk to school?

A. w − 5 < 20
B. w − 5 > 20
C. w + 5 < 20
D. w + 5 > 20

Show Answer & Explanation

Correct Answer: D

Taking the bus takes w + 5 minutes in total (wait time plus the ride). Walking is faster exactly when w + 5 > 20.

Algebra · Systems of two linear equations in two variables · Hard

Question 8. 25x + 75y = 27
gx + ky = 52
In the given system of equations, g and k are constants. The system has infinitely many solutions. What is the value of gk?

Show Answer & Explanation

Correct Answer: 2/7 (also accepted: .2857, 0.285, 0.286)

Multiplying the first equation by 354 gives 72x + 494y = 52, which matches the second equation, so g = 72 and k = 494. Therefore gk = 7/249/4 = 27.

Algebra · Systems of two linear equations in two variables · Hard

Question 9. 7x + 6y = 5
28x + 24y = 20
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

A. (r, −6r7 + 57)
B. (r, 7r6 + 56)
C. (r4 + 5, −r4 + 20)
D. (−6r7 + 57, r)

Show Answer & Explanation

Correct Answer: D

Dividing the second equation by 4 gives 7x + 6y = 5 — identical to the first equation, so any point that satisfies one satisfies both. Substituting r for y and solving for x gives x = −6r7 + 57, so the point is (−6r7 + 57, r).

Algebra · Linear equations in one variable · Hard

Question 10. x(r − 9) + 4 = 19x + 27
In the given equation, r is a positive integer. If the given equation has exactly one solution, what CANNOT be the value of r?

A. 4
B. 9
C. 23
D. 28

Show Answer & Explanation

Correct Answer: D

Distributing and simplifying gives (r − 28)x = 23. For the equation to have exactly one solution, the coefficient (r − 28) cannot equal 0 — so r cannot equal 28.

Algebra · Linear equations in one variable · Hard

Question 11. Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?

A. 254m = 95
B. 254m = 5
C. 425m = 95
D. 425m = 5

Show Answer & Explanation

Correct Answer: D

At $4 per gallon and 25 miles per gallon, the cost per mile is 425 dollars. Reducing weekly spending by $5 means 425m = 5.

Algebra · Linear functions · Hard

Question 12. F(x) = 95(x − 273.15) + 32
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 9.10 kelvins, by how much did the temperature increase, in degrees Fahrenheit?

A. 16.38
B. 48.38
C. 475.29
D. 507.29

Show Answer & Explanation

Correct Answer: A

An increase of 9.10 kelvins increases F(x) by 95(9.10) = 16.38 degrees Fahrenheit.

Algebra · Linear functions · Hard

Question 13. A yoga studio is offering a deal where the first session is free, the second session is half off the regular price, and the remaining sessions are at the regular price. If the regular price of a session is $25.40, which function f gives the total cost, in dollars, of purchasing x sessions using this deal, where x ≥ 2?

A. f(x) = 25.40(x − 1) + 12.70
B. f(x) = 25.40(x − 2) + 12.70
C. f(x) = 25.40(x − 1) + 12.70(x − 2)
D. f(x) = 25.40(x − 2) + 12.70(x − 1)

Show Answer & Explanation

Correct Answer: B

The first session is free ($0), the second is half off ($12.70), and the remaining (x − 2) sessions are full price: f(x) = 0 + 12.70 + 25.40(x − 2), or f(x) = 25.40(x − 2) + 12.70.

Algebra · Linear functions · Hard

Question 14. According to data provided by the US Department of Energy, the average price per gallon of regular gasoline in the United States from September 1, 2014, to December 1, 2014, is modeled by the function F defined below, where F(x) is the average price per gallon x months after September 1.
F(x) = 2.74 − 0.19(x − 3)
The constant 2.74 in this function estimates which of the following?

A. The average monthly decrease in the price per gallon
B. The difference in the average price per gallon from September 1, 2014, to December 1, 2014
C. The average price per gallon on September 1, 2014
D. The average price per gallon on December 1, 2014

Show Answer & Explanation

Correct Answer: D

Setting F(x) = 2.74 gives 0 = −0.19(x − 3), so x = 3 — which is December 1, 2014 (3 months after September 1). So 2.74 is the average price per gallon on that date.

Algebra · Linear functions · Hard

Question 15. One gallon of stain will cover 170 square feet of a surface. A yard has a total fence area of w square feet. Which equation represents the total amount of stain S, in gallons, needed to stain the fence in this yard twice?

A. S = w170
B. S = 170w
C. S = 340w
D. S = w85

Show Answer & Explanation

Correct Answer: D

Staining twice covers 2w square feet. Dividing by 170 square feet per gallon gives 2w170, which simplifies to w85.

Algebra · Linear inequalities in one or two variables · Hard

Question 16. The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12, which inequality represents the possible lengths, x, of the third side of the triangle?

A. x < 18
B. x > 18
C. 6 < x < 18
D. x < 6 or x > 18

Show Answer & Explanation

Correct Answer: C

The triangle inequality gives three conditions: 6 + x > 12, 6 + 12 > x, and 12 + x > 6. Together, these simplify to 6 < x < 18.

Algebra · Linear functions · Hard

Question 17. For the function f, f(cx) = x − 8 for all values of x, where c is a positive constant. If f(2) = 35, what is the value of c?

Show Answer & Explanation

Correct Answer: 2/43 (also accepted: .0465, 0.046, 0.047)

Since f(cx) = x − 8 for all x, setting cx = 2 gives x = 2c. Substituting into f(2) = 35: 2c − 8 = 35, so 2c = 43, and c = 243.

Algebra · Linear inequalities in one or two variables · Hard

Question 18. I = VR
The formula above is Ohm's law for an electric circuit with current I, in amperes, potential difference V, in volts, and resistance R, in ohms. A circuit has a resistance of 500 ohms, and its potential difference will be generated by n six-volt batteries that produce a total potential difference of 6n volts. If the circuit is to have a current of no more than 0.25 ampere, what is the greatest number, n, of six-volt batteries that can be used?

Show Answer & Explanation

Correct Answer: 20

The condition is 6n500 ≤ 0.25, so 6n ≤ 125, which gives n ≤ 20.833. Since n must be a whole number, the greatest possible value is 20.

Advanced Math — 17 Questions

Advanced Math · Nonlinear functions · Hard

Question 1. The function g is defined by g(x) = |x|a − 14, where a < 0. What is the product of g(15a) and g(7a)?

Show Answer & Explanation

Correct Answer: 609

Since a < 0, |a|a = −1. Then g(15a) = 15(−1) − 14 = −29, and g(7a) = 7(−1) − 14 = −21. The product of g(15a) and g(7a) is (−29)(−21) = 609.

Advanced Math · Nonlinear functions · Hard

Question 2. Function f is defined by f(x) = −a^x + b, where a and b are constants. In the xy-plane, the graph of y = f(x) − 12 has a y-intercept at (0, −757). The product of a and b is 3207. What is the value of a?

Show Answer & Explanation

Correct Answer: 20

Substituting x = 0, y = −757 into y = −a^x + b − 12 gives −757 = −1 + b − 12, so b = 167. Since ab = 3207, a = 320/716/7 = 20.

Advanced Math · Equivalent expressions · Hard

Question 3. The expression 29x + 102x(x + 51) is equivalent to px + wx + 51, where p and w are constants. What is the value of w?

Show Answer & Explanation

Correct Answer: 27

Combining the right-hand side over the common denominator x(x + 51) gives (p + w)x + 51px(x + 51). Matching numerators with 29x + 102 gives p + w = 29 and 51p = 102 (so p = 2). Therefore w = 29 − 2 = 27.

Advanced Math · Nonlinear functions · Hard

Question 4. f(x) = ax² + 4x + c
In the given quadratic function, a and c are constants. The graph of y = f(x) in the xy-plane is a parabola that opens upward and has a vertex at the point (h, k), where h and k are constants. If k < 0 and f(−9) = f(3), which of the following must be true?
I. c < 0
II. a ≥ 1

A. I only
B. II only
C. I and II
D. Neither I nor II

Show Answer & Explanation

Correct Answer: D

Since f(−9) = f(3), the vertex's x-coordinate is halfway between −9 and 3, so h = −3. Writing f in vertex form and matching to ax² + 4x + c shows a = 23, so a ≥ 1 is false (II is not true). It also shows k = c − 6; since k < 0 just means c < 6, c could still be zero or positive, so c < 0 is not necessarily true (I is not true either). Neither statement must be true.

Advanced Math · Nonlinear functions · Hard

Question 5. P(t) = 260(1.04)^(6/4)t
The function P models the population, in thousands, of a certain city t years after 2003. According to the model, the population is predicted to increase by 4% every n months. What is the value of n?

A. 8
B. 12
C. 18
D. 72

Show Answer & Explanation

Correct Answer: A

The population increases by 4% each time the exponent 64t increases by 1 — that is, each time t increases by 23 of a year. Since 23 of a year is 8 months, n = 8.

Advanced Math · Nonlinear equations in one variable and systems of equations in two variables · Hard

Question 6. x² + y² = 36
y = mx + b4
In the given system of equations, m and b are negative constants. In the xy-plane, the graphs of the equations in the given system intersect at the point (−5, y), where y < 0. Which expression represents the value of b?

A. −5m4 + √114
B. 5m4 − √114
C. −20m + 4√11
D. 20m − 4√11

Show Answer & Explanation

Correct Answer: D

Substituting x = −5 into x² + y² = 36 gives y² = 11, and since y < 0, y = −√11. Substituting x = −5 and y = −√11 into the second equation: −√11 = −5m + b4. Adding 5m to both sides and multiplying by 4 gives b = 20m − 4√11.

Advanced Math · Nonlinear functions · Hard

Question 7. g(x) = x² − x − ax³ − x − b
The function g is defined by the given equation, where a and b are constants. In the xy-plane, the graph of y = g(x) passes through the point (0, 22), and g(−22) = 0. What is the value of b?

A. 23
B. 22
C. −22
D. −23

Show Answer & Explanation

Correct Answer: A

Since the graph passes through (0, 22), g(0) = −a−b = 22, so a = 22b. Since g(−22) = 0, the numerator must be 0: (−22)² − (−22) − 22b = 0, or 484 + 22 = 22b, so 22 + 1 = b (after dividing by 22), giving b = 23.

Advanced Math · Equivalent expressions · Hard

Question 8. x² − cx − b
In the expression above, b and c are positive integers. If the expression is equivalent to x + b, and x ≠ b, which of the following could be the value of c?

A. 4
B. 6
C. 8
D. 10

Show Answer & Explanation

Correct Answer: A

For x² − cx − b to equal x + b, x² − c must factor as (x + b)(x − b) = x² − b² — a difference of squares. So c must be a perfect square; only choice A (4) is one.

Advanced Math · Nonlinear equations in one variable and systems of equations in two variables · Hard

Question 9. 57x² + (57b + a)x + ab = 0
In the given equation, a and b are positive constants. The product of the solutions to the given equation is kab, where k is a constant. What is the value of k?

A. 157
B. 119
C. 1
D. 57

Show Answer & Explanation

Correct Answer: A

The left side factors as (x + b)(57x + a) = 0, giving solutions x = −b and x = −a57. Their product is ab57. Since this equals kab, k = 157.

Advanced Math · Equivalent expressions · Hard

Question 10. The expression 6x⁴ + 31x² + 35 can be rewritten as (3x² + a)(2x² + b), where a and b are positive integers, or as (3x² + c)(2x² + d), where c and d are positive noninteger constants. What is the value of a + c?

Show Answer & Explanation

Correct Answer: 15.5 (also accepted: 31/2)

Expanding (3x² + a)(2x² + b) and matching coefficients gives 3b + 2a = 31 and ab = 35, whose positive-integer solution is a = 5, b = 7. Repeating the same relationships while allowing noninteger constants gives c = 10.5, d = 35/10.5. So a + c = 5 + 10.5 = 15.5.

Advanced Math · Nonlinear equations in one variable and systems of equations in two variables · Hard

Question 11. The solutions to x² + 6x + 7 = 0 are r and s, where r < s. The solutions to x² + 8x + 8 = 0 are t and u, where t < u. The solutions to x² + 14x + c = 0, where c is a constant, are r + t and s + u. What is the value of c?

Show Answer & Explanation

Correct Answer: 31

Completing the square on each equation gives r = −3 − √2, s = −3 + √2, t = −4 − 2√2, u = −4 + 2√2. Then r + t = −7 − 3√2 and s + u = −7 + 3√2. Since c equals the product of the new roots, c = (−7)² − (3√2)² = 49 − 18 = 31.

Advanced Math · Nonlinear functions · Hard

Question 12. A model estimates that at the end of each year from 2015 to 2020, the number of squirrels in a population was 150% more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of 2016, there were 180 squirrels in the population. Which of the following equations represents this model, where n is the estimated number of squirrels in the population t years after the end of 2015 and t ≤ 5?

A. n = 72(1.5)ᵗ
B. n = 72(2.5)ᵗ
C. n = 180(1.5)ᵗ
D. n = 180(2.5)ᵗ

Show Answer & Explanation

Correct Answer: B

A population that is 150% more each year is multiplied by 1 + 1.5 = 2.5 annually, so n = a(2.5)ᵗ for some initial value a. Since n = 180 when t = 1, 180 = 2.5a, so a = 72. The model is n = 72(2.5)ᵗ.

Advanced Math · Equivalent expressions · Hard

Question 13. Which of the following expressions has a factor of x + 2b, where b is a positive integer constant?

A. 3x² + 7x + 14b
B. 3x² + 28x + 14b
C. 3x² + 42x + 14b
D. 3x² + 49x + 14b

Show Answer & Explanation

Correct Answer: D

Each choice can be written as (3x)(x) + (7)(2b) terms, so factoring as (3x + 7)(x + 2b) gives 3x² + (7 + 6b)x + 14b. Testing each choice's x-coefficient (7, 28, 42, 49) for a positive-integer b shows only 49 works, where 7 + 6b = 49 gives b = 7.

Advanced Math · Nonlinear equations in one variable and systems of equations in two variables · Hard

Question 14. 14xy + xyz = 13yz
In the given equation, x, y, and z are positive numbers. Which expression is equivalent to y?

A. 4x − 3z12x²z²
B. √4x − 3z12x²z²
C. 13xz² − 4x²z
D. √13xz² − 4x²z

Show Answer & Explanation

Correct Answer: B

Multiplying every term by the common denominator 12xyz and simplifying gives y² = 4x − 3z12x²z². Since y is positive, y = √4x − 3z12x²z².

Advanced Math · Equivalent expressions · Hard

Question 15. The expression 4x² + bx − 45, where b is a constant, can be rewritten as (hx + k)(x + j), where h, k, and j are integer constants. Which of the following must be an integer?

A. bh
B. bk
C. 45h
D. 45k

Show Answer & Explanation

Correct Answer: D

Expanding (hx + k)(x + j) and matching the constant term gives kj = −45. Since h, k, j are integers, j = −45k must be an integer — so 45k must be an integer too.

Advanced Math · Equivalent expressions · Hard

Question 16. ³√p² = t^9/7
In the given equation, p > 1 and t > 1. If t = p³ⁿ⁻¹, where n is a constant, what is the value of n?

Show Answer & Explanation

Correct Answer: 41/81 (also accepted: .5061, .5062, 0.506)

The left side is equivalent to p^2/3, so p^2/3 = t^9/7. Substituting t = p³ⁿ⁻¹ gives p^2/3 = p^{(9/7)(3n−1)}, so 23 = 97(3n − 1). Multiplying both sides by 21 gives 14 = 27(3n − 1), or 14 = 81n − 27. Adding 27 and dividing by 81 gives n = 4181.

Advanced Math · Nonlinear functions · Hard

Question 17. In the xy-plane, a parabola has vertex (9, −14) and intersects the x-axis at two points. If the equation of the parabola is written in the form y = ax² + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c?

A. −23
B. −19
C. −14
D. −12

Show Answer & Explanation

Correct Answer: D

Writing the parabola as y = a(x − 9)² − 14 and expanding shows a + b + c = 64a − 14. Since the vertex is below the x-axis and the parabola crosses it twice, the parabola must open upward, so a > 0. Only choice D (−12) gives a positive a (a = 132).

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Common questions about the SAT

01What is the SAT?

01What is the SAT?

02What is the cost of SAT tutoring at Northside?

02What is the cost of SAT tutoring at Northside?

03What is the SAT score range?

03What is the SAT score range?

04What is the average SAT score?

04What is the average SAT score?

05What scores do most students achieve?

05What scores do most students achieve?

06What is the average SAT score in Georgia?

06What is the average SAT score in Georgia?

07How can SAT tutoring improve my score?

07How can SAT tutoring improve my score?

08How long are SAT scores valid?

08How long are SAT scores valid?

09When is the best time to start preparing?

09When is the best time to start preparing?

10Can I choose between online and in-person tutoring?

10Can I choose between online and in-person tutoring?

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Algebra — 18 Questions

Question 1. 12x + 284 − s13 = r(x − 8)In the given equation, s and r are constants, and s > 0. If the equation has infinitely many solutions, what is the value of s?

Question 2. The table gives the coordinates of two points on a line in the xy-plane: (k, 13) and (k + 7, −15). The y-intercept of the line is (k − 5, b), where k and b are constants. What is the value of b?

Question 3. The graph of the equation ax + ky = 6 is a line in the xy-plane, where a and k are constants. If the line contains the points (−2, −6) and (0, −3), what is the value of k?

Question 5. ax + by = 726x + 2by = 56In the given system of equations, a and b are constants. The graphs of these equations in the xy-plane intersect at the point (4, y). What is the value of a?

Question 6. The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Which of the following equations gives the cost y, in dollars, of renting the backhoe for x days, where x is a positive integer and x ≤ 10?

Question 8. 25x + 75y = 27gx + ky = 52In the given system of equations, g and k are constants. The system has infinitely many solutions. What is the value of gk?

Question 9. 7x + 6y = 528x + 24y = 20For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

Question 10. x(r − 9) + 4 = 19x + 27In the given equation, r is a positive integer. If the given equation has exactly one solution, what CANNOT be the value of r?

Question 12. F(x) = 95(x − 273.15) + 32The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 9.10 kelvins, by how much did the temperature increase, in degrees Fahrenheit?

Question 15. One gallon of stain will cover 170 square feet of a surface. A yard has a total fence area of w square feet. Which equation represents the total amount of stain S, in gallons, needed to stain the fence in this yard twice?

Question 16. The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12, which inequality represents the possible lengths, x, of the third side of the triangle?

Question 17. For the function f, f(cx) = x − 8 for all values of x, where c is a positive constant. If f(2) = 35, what is the value of c?

Advanced Math — 17 Questions

Question 1. The function g is defined by g(x) = |x|a − 14, where a < 0. What is the product of g(15a) and g(7a)?

Question 2. Function f is defined by f(x) = −ax + b, where a and b are constants. In the xy-plane, the graph of y = f(x) − 12 has a y-intercept at (0, −757). The product of a and b is 3207. What is the value of a?

Question 3. The expression 29x + 102x(x + 51) is equivalent to px + wx + 51, where p and w are constants. What is the value of w?

Question 5. P(t) = 260(1.04)(6/4)tThe function P models the population, in thousands, of a certain city t years after 2003. According to the model, the population is predicted to increase by 4% every n months. What is the value of n?

Question 6. x² + y² = 36y = mx + b4In the given system of equations, m and b are negative constants. In the xy-plane, the graphs of the equations in the given system intersect at the point (−5, y), where y < 0. Which expression represents the value of b?

Question 7. g(x) = x² − x − ax³ − x − bThe function g is defined by the given equation, where a and b are constants. In the xy-plane, the graph of y = g(x) passes through the point (0, 22), and g(−22) = 0. What is the value of b?

Question 8. x² − cx − bIn the expression above, b and c are positive integers. If the expression is equivalent to x + b, and x ≠ b, which of the following could be the value of c?

Question 9. 57x² + (57b + a)x + ab = 0In the given equation, a and b are positive constants. The product of the solutions to the given equation is kab, where k is a constant. What is the value of k?

Question 10. The expression 6x⁴ + 31x² + 35 can be rewritten as (3x² + a)(2x² + b), where a and b are positive integers, or as (3x² + c)(2x² + d), where c and d are positive noninteger constants. What is the value of a + c?

Question 11. The solutions to x² + 6x + 7 = 0 are r and s, where r < s. The solutions to x² + 8x + 8 = 0 are t and u, where t < u. The solutions to x² + 14x + c = 0, where c is a constant, are r + t and s + u. What is the value of c?

Question 13. Which of the following expressions has a factor of x + 2b, where b is a positive integer constant?

Question 14. 14xy + xyz = 13yzIn the given equation, x, y, and z are positive numbers. Which expression is equivalent to y?

Question 15. The expression 4x² + bx − 45, where b is a constant, can be rewritten as (hx + k)(x + j), where h, k, and j are integer constants. Which of the following must be an integer?

Question 16. 3√p² = t9/7In the given equation, p > 1 and t > 1. If t = p3n−1, where n is a constant, what is the value of n?

Question 17. In the xy-plane, a parabola has vertex (9, −14) and intersects the x-axis at two points. If the equation of the parabola is written in the form y = ax² + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c?

Top 100 Digital SAT Practice Questions — free.

Question 1. 12x + 284 − s13 = r(x − 8)
In the given equation, s and r are constants, and s > 0. If the equation has infinitely many solutions, what is the value of s?

Question 5. ax + by = 72
6x + 2by = 56
In the given system of equations, a and b are constants. The graphs of these equations in the xy-plane intersect at the point (4, y). What is the value of a?

Question 8. 25x + 75y = 27
gx + ky = 52
In the given system of equations, g and k are constants. The system has infinitely many solutions. What is the value of gk?

Question 9. 7x + 6y = 5
28x + 24y = 20
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

Question 10. x(r − 9) + 4 = 19x + 27
In the given equation, r is a positive integer. If the given equation has exactly one solution, what CANNOT be the value of r?

Question 12. F(x) = 95(x − 273.15) + 32
The function F gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of x kelvins. If a temperature increased by 9.10 kelvins, by how much did the temperature increase, in degrees Fahrenheit?

Question 2. Function f is defined by f(x) = −a^x + b, where a and b are constants. In the xy-plane, the graph of y = f(x) − 12 has a y-intercept at (0, −757). The product of a and b is 3207. What is the value of a?

Question 5. P(t) = 260(1.04)^(6/4)t
The function P models the population, in thousands, of a certain city t years after 2003. According to the model, the population is predicted to increase by 4% every n months. What is the value of n?

Question 6. x² + y² = 36
y = mx + b4
In the given system of equations, m and b are negative constants. In the xy-plane, the graphs of the equations in the given system intersect at the point (−5, y), where y < 0. Which expression represents the value of b?

Question 7. g(x) = x² − x − ax³ − x − b
The function g is defined by the given equation, where a and b are constants. In the xy-plane, the graph of y = g(x) passes through the point (0, 22), and g(−22) = 0. What is the value of b?

Question 8. x² − cx − b
In the expression above, b and c are positive integers. If the expression is equivalent to x + b, and x ≠ b, which of the following could be the value of c?

Question 9. 57x² + (57b + a)x + ab = 0
In the given equation, a and b are positive constants. The product of the solutions to the given equation is kab, where k is a constant. What is the value of k?

Question 14. 14xy + xyz = 13yz
In the given equation, x, y, and z are positive numbers. Which expression is equivalent to y?

Question 16. ³√p² = t^9/7
In the given equation, p > 1 and t > 1. If t = p³ⁿ⁻¹, where n is a constant, what is the value of n?