Systems of linear equations are one of the highest-yield topics in the "Algebra" domain on Digital SAT Math, and the College Board tests them in several distinct ways: solving a system algebraically for a specific variable or expression, recognizing when a system has no solution or infinitely many solutions, translating a word problem into two equations, and reading a graph to find where two lines intersect. A large share of these questions can be solved quickly using elimination rather than full substitution — especially the "what is the value of 7x + 7y" style questions, where adding or subtracting the equations directly gets you to the answer in one step.
The 23 problems below cover every version of this skill that shows up on the real exam, including several real-world setups (ticket sales, calorie counts, wedding seating, chocolate distribution) and the special cases of no solution and infinitely many solutions, which trip up even strong students. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. Pay close attention to the no-solution and infinite-solutions problems in particular — recognizing what makes a system parallel (no solution) versus identical (infinite solutions) is one of the most commonly misunderstood ideas in this topic. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
Which of the following is a graph of a system of equations with no solution?
A system of linear equations has no solution only when the two lines are parallel — same slope, different y-intercepts — so they never intersect.
Only one graph shows two straight lines that stay the same distance apart and never cross. The correct answer is B.
An aquarium sells two types of tickets. The standard ticket costs $20, and the premium ticket costs $40. One Tuesday, the aquarium sold 400 total tickets and collected $9,000. Which system of equations can be used to find the number of standard tickets s and premium tickets p sold?
The total number of tickets sold gives one equation: s + p = 400.
The total money collected gives the second equation, using each ticket's price: 20s + 40p = 9,000. The correct answer is A.
The equation ax = 3, where a is a constant, will have no solution for which of the following values of a?
To solve ax = 3 for x, divide both sides by a: x = 3/a. This only fails when a = 0, since division by zero is undefined.
When a = 0, the equation becomes 0 = 3, which is never true — no value of x can solve it. The correct answer is A.
The system of equations (1/3)y = 6 and x − (1/3)y = 2 has solution (x, y). What is the value of x?
From the first equation, (1/3)y = 6, so y = 18.
Substitute into the second equation: x − (1/3)(18) = 2, so x − 6 = 2, giving x = 8. The correct answer is C.
Marcus sells only hats and scarves on his website. Hats sell for $15 each, and scarves sell for $10 each. If Marcus sold 30 pieces of clothing and his sales totaled $400, how many scarves did Marcus sell?
Let h = number of hats and s = number of scarves. From the total items: h + s = 30, so h = 30 − s.
From total sales: 15h + 10s = 400. Substitute: 15(30 − s) + 10s = 400, so 450 − 15s + 10s = 400, giving 450 − 5s = 400, so 5s = 50 and s = 10.
In the equation 12x − 4x(c − 2) = 3x, c is a constant. If the equation has infinitely many solutions, what is the value of c?
Expand the left side: 12x − 4xc + 8x = 3x, which simplifies to 20x − 4xc = 3x, or x(20 − 4c) = 3x.
For this to be true for every value of x (infinitely many solutions), the coefficients on both sides must match exactly: 20 − 4c = 3, so 4c = 17, giving c = 17/4. The correct answer is D.
Which ordered pair (x, y) satisfies the system of equations x = y − 5 and x/2 + 4y = 2?
Substitute x = y − 5 into the second equation: (y − 5)/2 + 4y = 2.
Multiply everything by 2: (y − 5) + 8y = 4, so 9y − 5 = 4, giving y = 1. Then x = 1 − 5 = −4. The correct answer is B.
In the system of equations x/y = 4 and 3(y + 2) = x, what is the value of y?
From the first equation, x/y = 4, so x = 4y.
Substitute into the second equation: 3(y + 2) = 4y, so 3y + 6 = 4y, giving y = 6. The correct answer is B.
Based on the system of equations 2x + 5y = 800 and 5x + 2y = 1,500, what is the value of 7x + 7y?
Rather than solving for x and y separately, add the two equations directly: (2x + 5y) + (5x + 2y) = 800 + 1,500.
The left side combines to 7x + 7y, and the right side is 2,300. So 7x + 7y = 2,300.
If (x, y) is a solution to the system 4(x + y) = 56 and x/2 = 5, what is the value of y?
From the second equation, x/2 = 5, so x = 10.
From the first equation, 4(x + y) = 56, so x + y = 14. Substitute x = 10: 10 + y = 14, giving y = 4. The correct answer is B.
If 3w + 5t = 21 and 5w − t = 7, what is the value of 4w + 2t?
Solve for t using the second equation: t = 5w − 7. Substitute into the first equation: 3w + 5(5w − 7) = 21, so 3w + 25w − 35 = 21, giving 28w = 56 and w = 2.
Then t = 5(2) − 7 = 3. So 4w + 2t = 4(2) + 2(3) = 8 + 6 = 14. The correct answer is A.
At a food truck, each empanada has 40 more calories than each taco. If 1 empanada and 8 tacos have a total of 1,570 calories, how many calories does an empanada have?
Let t = calories per taco. Then the empanada has t + 40 calories. The total is (t + 40) + 8t = 1,570.
Combine like terms: 9t + 40 = 1,570, so 9t = 1,530, giving t = 170. The empanada has 170 + 40 = 210 calories.
The solution to the system of equations x − 4y = 24 and 0.25x + y = 4 is (x, y). What is the value of x?
From the first equation, x = 24 + 4y. Substitute into the second equation: 0.25(24 + 4y) + y = 4.
Distribute: 6 + y + y = 4, so 6 + 2y = 4, giving 2y = −2 and y = −1. Then x = 24 + 4(−1) = 20. The correct answer is C.
If (x, y) satisfies the system −x + y = −4.75 and x + 5y = 11.75, what is the value of y?
Add the two equations directly — the x-terms cancel: (−x + y) + (x + 5y) = −4.75 + 11.75.
That gives 6y = 7, so y = 7/6 (approximately 1.17).
In the xy-plane, the point (p, r) lies on the line with equation y = 2x + b, where b is a constant. The point with coordinates (4p, 3r) lies on the line with equation y = 3x + b. If p ≠ 0, what is the value of r/p?
Since (p, r) is on the first line: r = 2p + b.
Since (4p, 3r) is on the second line: 3r = 3(4p) + b = 12p + b. Substitute b = r − 2p from the first equation: 3r = 12p + r − 2p = 10p + r. Subtract r from both sides: 2r = 10p, so r/p = 5. The correct answer is A.
The equation 7x − 4 = a(x − b), where a and b are constants, has no solutions. Which of the following must be true? I. a = 7 II. b = 4 III. b ≠ 4/7
Expand the right side: 7x − 4 = ax − ab. For the equation to have no solution, the x-coefficients must match (so the two sides represent parallel, non-identical lines): a = 7. That confirms statement I.
With a = 7, the constant terms must differ: −4 ≠ −ab, so −4 ≠ −7b, meaning b ≠ 4/7. That confirms statement III, but statement II (b = 4) is not required. The correct answer is D.
In a wedding hall, guests are seated at tables. If 6 guests are assigned to each table, 3 additional tables will be needed to seat all of the guests. If 12 guests are assigned to each table, 3 tables will not be used. How many guests will be attending the wedding reception?
Let T be the number of tables normally available and G be the number of guests. Seating 6 per table needs 3 extra tables: 6(T + 3) = G. Seating 12 per table leaves 3 tables unused: 12(T − 3) = G.
Set them equal: 6T + 18 = 12T − 36, so 54 = 6T, giving T = 9. Then G = 6(9 + 3) = 6(12) = 72 guests.
The system of equations (2/3)x − (4/5)y = 10 and ax − by = 15 has no solutions. If a and b are constants, what is the value of a/b?
For a system to have no solution, the two equations must represent parallel lines: the coefficients of x and y must be proportional to each other, but not proportional to the constant term.
Set up the ratio of coefficients: a / (2/3) = b / (4/5), so a/b = (2/3) / (4/5) = (2/3) × (5/4) = 10/12 = 5/6.
In the xy-plane, the lines that correspond to the system of equations (1/4)y = 21/20 − (1/5)x and 8y = 9x intersect at the point (a, b). What is the value of a/b?
Clear fractions in the first equation by multiplying by 20: 5y = 21 − 4x, or 4x + 5y = 21. The second equation, 8y = 9x, gives x = 8y/9.
Substitute into the first equation: 4(8y/9) + 5y = 21, so 32y/9 + 45y/9 = 21, giving 77y/9 = 21, so y = 189/77 = 27/11. Then x = 8(27/11)/9 = 24/11. So a/b = x/y = (24/11) / (27/11) = 24/27 = 8/9.
The score on a history test is obtained by subtracting the number of incorrect answers from 4 times the number of correct answers. If a student answered all 25 questions and obtained a score of 65, how many questions did the student answer correctly?
Let c = correct answers and i = incorrect answers. Since every question was answered: c + i = 25. The scoring rule gives: 4c − i = 65.
Add the two equations to eliminate i: (c + i) + (4c − i) = 25 + 65, so 5c = 90, giving c = 18.
In one month, Dan and John spent a combined 500 hours writing a math book. If Dan spent 50 more hours writing the book than John did, how many hours did John spend writing the book?
Let J = John's hours. Then Dan's hours = J + 50. Together: J + (J + 50) = 500.
Combine like terms: 2J + 50 = 500, so 2J = 450, giving J = 225 hours.
In the system of equations 2x + y = a and −x + 2y = 4, a is a constant. What is the y-value of the solution to the system in terms of a?
Multiply the first equation by 2: 4x + 2y = 2a. Subtract the second equation, −x + 2y = 4, from this: (4x + 2y) − (−x + 2y) = 2a − 4, giving 5x = 2a − 4, so x = (2a − 4)/5.
Substitute back into −x + 2y = 4: 2y = 4 + x = 4 + (2a − 4)/5 = (20 + 2a − 4)/5 = (2a + 16)/5. Dividing by 2: y = (a + 8)/5. The correct answer is A.
Ms. Morrissey has a bag containing n pieces of chocolate to distribute to the students in her second grade class. If she gives each student 2 pieces of chocolate, she will have 6 pieces left over. In order to give each student 3 pieces of chocolate, she will need an additional 16 pieces. How many students are in the class?
Let s = number of students. Giving 2 pieces each with 6 left over: n = 2s + 6. Giving 3 pieces each requires 16 more pieces than she has: n + 16 = 3s, so n = 3s − 16.
Set the two expressions for n equal: 2s + 6 = 3s − 16, so 22 = s. The correct answer is D.
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