These are 5 hard SAT algebra questions from our question bank, all at the top of our difficulty scale. Algebra is about 35% of the Math section, 13 to 15 questions, and it covers linear equations, linear functions, systems and inequalities. The hard ones rarely need new math. They add an unknown constant, extra information, or a word problem you must translate. Try each one, then open the solution.
Question 1 (Hard, Lines and Linear Relationships)
- A. -1
- B. 4
- C. 6
- D. 8
Show answer and solution
Answer: D. 8
First find c from p. The slope of p is 6, so (34 − (−2)) / (5 − c) = 6. That gives 36 = 6(5 − c), so c = −1. Then t passes through (−1, −4) and (6, 52), so its slope is (52 − (−4)) / (6 − (−1)) = 56 / 7 = 8.
Tip: When two functions share a constant, solve for it with the function you know more about.
Question 2 (Hard, Interpreting Linear Equations)
19.52x + 24.25y = 583
Odalys ordered mulch and river rock, which cost a total of $583, for her home. The given equation represents the relationship between the number of cubic yards of mulch, x, and the number of tons of river rock, y, Odalys ordered. how much more, in dollars, did a ton of river rock cost Odalys than a cubic yard of mulch?
- A. 4.62
- B. 4.73
- C. 4.78
- D. 4.85
Show answer and solution
Answer: B. 4.73
In 19.52x + 24.25y = 583, each coefficient is a price: 19.52 dollars per cubic yard of mulch and 24.25 dollars per ton of river rock. 24.25 − 19.52 = 4.73.
Tip: In a cost equation, the coefficient next to each variable is the price of one unit of that item.
Question 3 (Hard, Linear Inequalities)
Students in a science lab are working in groups to build both a small and a large electrical circuit. A large circuit uses 4 resistors and 2 capacitors, and a small circuit uses 3 resistors and 1 capacitor. There are 100 resistors and 70 capacitors available, and each group must have enough resistors and capacitors to make one large and one small circuit. What is the maximum number of groups that could work on this lab project?
- A. 11
- B. 12
- C. 13
- D. 14
Show answer and solution
Answer: D. 14
Each group builds one large and one small circuit: 4 + 3 = 7 resistors and 2 + 1 = 3 capacitors. Resistors allow 100 / 7 ≈ 14.3 groups, and capacitors allow 70 / 3 ≈ 23.3. The smaller limit wins, so 14 groups.
Tip: With two resources, find the limit from each one. The smaller limit is the answer.
Review: Linear Inequalities notes
Want more at this level?
Targeted Practice builds sets from the skills you miss.
Question 4 (Hard, Lines and Linear Relationships)
| x | y |
| k | 13 |
| k+7 | -15 |
The table gives the coordinates of two points on a line in the xy plane. The y-intercept of the line is (k-5, b), where k and b are constants. What is the value of b?
- A. 17
- B. 21
- C. 26
- D. 33
Show answer and solution
Answer: D. 33
The slope is (−15 − 13) / ((k + 7) − k) = −28 / 7 = −4. A y-intercept always has x = 0, so k − 5 = 0 and k = 5. The line passes through (5, 13), so b = 13 + 4(5) = 33.
Tip: Any y-intercept has x = 0. That one fact often unlocks the unknown constant.
Question 5 (Hard, Linear vs. Exponential Models)
Everest Education Center currently has 2000 students and expects growth for the next 5 years. The institution board is comparing two different growth projections. One model predicts that each year there will be 30 more students than the previous year. The other model predicts that each year there will be 1% more students than the previous year. To the nearest whole number, what is the difference in the projection of the total number of students these models predict after 5 years?
- A. 10
- B. 20
- C. 29
- D. 48
Show answer and solution
Answer: D. 48
The linear model adds 30 a year: 2000 + 5(30) = 2150. The exponential model grows 1% a year: 2000(1.01)5 ≈ 2102.02. The difference is about 48.
Tip: Adding the same amount each year is linear. Multiplying by the same factor each year is exponential.
Review: Linear vs. Exponential Models notes
How to get better
- Learn the idea behind each solution. Each tip above is a pattern that shows up on many hard questions.
- Read the notes linked in each solution, then drill the skill in Targeted Practice.
- Watch it explained in the free Advanced Math video series with Mr. O.
- Test yourself under time with a free full-length practice test.
Want more? Try these hard SAT math questions and hard SAT geometry questions.
Frequently asked questions
How many algebra questions are on the SAT?
Algebra is about 35% of the Math section, 13 to 15 of the 44 questions.
What algebra is on the SAT?
Linear equations in one and two variables, linear functions, systems of two linear equations, and linear inequalities.
Is algebra the hardest part of SAT Math?
Most hard questions come from Algebra and Advanced Math, usually with an extra step or an unknown constant.
Hard SAT Math practice
This guide is part of Digital SAT Vocabulary and Advanced Math: Resource Hub.
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Sources
- College Board, The Math Section: Overview.
Frequently Asked Questions
How many algebra questions are on the SAT?
Algebra is about 35% of the Math section, 13 to 15 of the 44 questions.
What algebra is on the SAT?
Linear equations in one and two variables, linear functions, systems of two linear equations, and linear inequalities.
Is algebra the hardest part of SAT Math?
Most hard questions come from Algebra and Advanced Math, usually with an extra step or an unknown constant.