Beyond just solving a system of equations for x and y, Digital SAT Math frequently tests a deeper layer of understanding: how many solutions a system has, what makes two equations represent the same line versus parallel lines, how to identify a missing equation that would create no solution or infinitely many solutions, and how to read an intersection point directly off two graphed lines. These "special case" questions trip up students who only practiced mechanical substitution and elimination, because they require recognizing the relationship between the coefficients of two equations rather than grinding through algebra.
The 31 problems below cover this full range, including several "which equation completes a no-solution system" problems, systems with three equations, a quadratic-linear system with no real solution, and elimination shortcuts for finding a target expression without solving for x and y individually. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. The key skill to master here is comparing coefficient ratios: if two equations have the same ratio of x-coefficient to y-coefficient but a different constant, the system has no solution; if every ratio matches including the constant, it has infinitely many. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
The solution to the system of equations 5x + 4y = 44 and 3x − 7y = 17 is (x, y). What is the value of 8x − 3y?
Multiply the first equation by 7 and the second by 4 to align the y-coefficients: 35x + 28y = 308 and 12x − 28y = 68. Adding these eliminates y: 47x = 376, so x = 8.
Substitute x = 8 into the first equation: 5(8) + 4y = 44, so 40 + 4y = 44, giving y = 1. Then 8x − 3y = 8(8) − 3(1) = 64 − 3 = 61.
2x − 2y = 2. One of the two linear equations in a system is given. The system has exactly one solution. Which equation could be the second equation in this system?
Simplify the given equation: x − y = 1. A system has exactly one solution only when the second equation has a different ratio of x-coefficient to y-coefficient than 1 : −1.
Options A, B, and D all simplify to the same ratio (1 : −1) as the original — making them either parallel (no solution) or the same line (infinite solutions). Option C has coefficients −10 and 8, giving a ratio of −10/8, which is different from 1/−1, so the lines intersect at exactly one point. The correct answer is C.
y = 18x + 25 and y = −14x − 7. What is the solution (x, y) to the given system of equations?
Since both equations equal y, set them equal to each other: 18x + 25 = −14x − 7.
Add 14x to both sides: 32x + 25 = −7, so 32x = −32, giving x = −1. Substitute back: y = 18(−1) + 25 = 7. The solution is (−1, 7). The correct answer is B.
3x + 5y = 8 and 9x + 15y = 24. 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?
Notice the second equation is exactly 3 times the first (3 × (3x+5y) = 9x+15y, and 3 × 8 = 24), so both equations describe the same line — there are infinitely many solutions, one for every value of y.
Let y = r and solve the first equation for x: 3x = 8 − 5r, so x = 8/3 − 5r/3 = −5r/3 + 8/3. That gives the point (−5r/3 + 8/3, r). The correct answer is C.
mx + ky = −83 and 2x + ky = 22. In the given system, m and k are constants. The graphs of these equations intersect at the point (5, y). What is the value of m?
Substitute x = 5 into the second equation to find ky: 2(5) + ky = 22, so 10 + ky = 22, giving ky = 12.
Substitute x = 5 into the first equation: 5m + ky = −83. Since ky = 12: 5m + 12 = −83, so 5m = −95, giving m = −19.
2x + y = 8 and ax + 3y = 24. In the given system of equations, a is a constant. The system has infinitely many solutions. What is the value of a?
For infinitely many solutions, every ratio between the two equations must match: the x-coefficients, y-coefficients, and constants must all scale by the same factor.
Comparing y-coefficients and constants: 3/1 = 3 and 24/8 = 3, so the scale factor is 3. Applying that to the x-coefficients: a/2 = 3, so a = 6.
2x = −4 and 2x + y = −2. The solution to the given system of equations is (x, y). What is the value of 4x + y?
From the first equation, x = −2. Substitute into the second equation: 2(−2) + y = −2, so −4 + y = −2, giving y = 2.
Then 4x + y = 4(−2) + 2 = −8 + 2 = −6. The correct answer is A.
A small business budgeted $1,000 to spend on printers and toner cartridges. The small business paid $300 for each printer and $50 for each toner cartridge. The entire budget was spent to purchase 2 printers and several toner cartridges. How many toner cartridges were purchased?
Two printers cost 2 × $300 = $600. That leaves $1,000 − $600 = $400 for toner cartridges.
Divide the remaining budget by the cost per cartridge: $400 ÷ $50 = 8 cartridges. The correct answer is B.
3x + y = 12 and x − 3y = 0. The solution to the given system is (x, y). What is the value of x?
From the second equation, x = 3y. Substitute into the first equation: 3(3y) + y = 12, so 9y + y = 12, giving 10y = 12 and y = 1.2.
Then x = 3(1.2) = 3.6.
4x − 16y = 8. One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?
Simplify the given equation by dividing by 4: x − 4y = 2. For no solution, the second equation needs the exact same coefficient ratio (1 to −4) but a different constant.
Option C, (1/2)x − 2y = 0, doubles to x − 4y = 0 — the same ratio as the original, but a different constant (0 instead of 2), making the lines parallel with no intersection. (Option D is identical to the original equation, giving infinite solutions instead.) The correct answer is C.
y = −(1/9)x and y = (1/7)x. The solution to the given system of equations is (x, y). What is the value of x?
Set the two expressions for y equal: −x/9 = x/7. Multiply both sides by 63 to clear denominators: −7x = 9x.
Add 7x to both sides: 0 = 16x, so x = 0. The correct answer is C.
3x − 4y = 5 and x = 7. What is the solution (x, y) to the given system of equations?
Substitute x = 7 into the first equation: 3(7) − 4y = 5, so 21 − 4y = 5.
Subtract 21 from both sides: −4y = −16, so y = 4. The solution is (7, 4). The correct answer is A.
4x − 16y = 8. One of the two equations in a system of linear equations is given. The system has no solution. Which of the following could be the second equation in this system?
This uses the same setup as the earlier problem: simplify 4x − 16y = 8 by dividing by 4 to get x − 4y = 2.
Option C doubles to x − 4y = 0, matching the coefficient ratio of the original equation but with a different constant — parallel lines, no solution. The correct answer is C.
Which system of linear equations has no solution?
Check each pair for the same coefficient ratio with a different constant. In option A, multiplying the first equation by −1 gives 2x − 8y = 5, identical to the second equation — infinitely many solutions, not none. In option C, multiplying the first equation by −3 gives −6x + 24y = −15, again identical to the second — also infinitely many.
In option D, multiplying the first equation by −2 gives 4x − 16y = −10, which has the same coefficients as the second equation but a different constant (−10 vs. 10) — parallel lines with no solution. The correct answer is D.
8x + 7y = 58 and 8x + 21y = 142. The solution to the system is (x, y). What is the value of 14y?
Since both equations share the same 8x term, subtract the first equation from the second to eliminate x directly: (8x + 21y) − (8x + 7y) = 142 − 58.
That gives 14y = 84 immediately — exactly the value the question asks for. The correct answer is C.
5x = 50y − 125. One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?
Divide the given equation by 5: x = 10y − 25. For no solution, the second equation needs the same relationship between x and y (coefficient of y equal to 10) but a different constant.
Option B, (1/5)x = 2y, multiplies out to x = 10y — the same coefficient of y (10) as the original, but a different constant (0 instead of −25), making the lines parallel with no intersection. (Options C and D are both identical to the original equation, giving infinite solutions instead.) The correct answer is B.
x + 4y = −16, 5x + 20y = 0, and 3x + 12y = 34 are graphed in the xy-plane. How many solutions (x, y) does the system of three equations have?
Multiply the first equation by 5: 5x + 20y = −80. But the second equation states 5x + 20y = 0. Since −80 ≠ 0, these two equations alone already contradict each other.
Because the first two equations can't be satisfied simultaneously, the full system of three has no solution at all. The correct answer is A.
x + 5y = −18, 6x + 30y = 0, and 4x + 20y = 28 are graphed in the xy-plane. How many solutions (x, y) does the system of equations have?
Multiply the first equation by 6: 6x + 30y = −108. But the second equation states 6x + 30y = 0. Since −108 ≠ 0, these two equations directly contradict each other.
Since the first two equations alone are already inconsistent, the full system has no solution. The correct answer is A.
y = 6x² − 36x + 56 and y + 3 = 0. How many solutions are there to the given system of equations?
The second equation gives y = −3. Substitute into the first equation: −3 = 6x² − 36x + 56, which rearranges to 6x² − 36x + 59 = 0.
Check the discriminant: b² − 4ac = (−36)² − 4(6)(59) = 1,296 − 1,416 = −120. A negative discriminant means there are no real solutions — the parabola never reaches y = −3. The correct answer is D.
2(9x) + 5(8y) = 284 and 2(9x) − 5(8y) = −356. The solution to the given system of equations is (x, y). What is the value of 9x + 8y?
Let A = 9x and B = 8y to simplify: 2A + 5B = 284 and 2A − 5B = −356. Adding these equations eliminates B: 4A = −72, so A = 9x = −18.
Subtracting the equations eliminates A instead: 10B = 640, so B = 8y = 64. Then 9x + 8y = −18 + 64 = 46. The correct answer is C.
Two customers bought the same kind of cheese and milk at a store. The first customer paid $12.45 for 1 pound of cheese and 2 gallons of milk. The second customer paid $19.70 for 3 pounds of cheese and 1 gallon of milk. What is the cost, in dollars, of 1 gallon of milk?
Let c = cost per pound of cheese and m = cost per gallon of milk. The two purchases give: c + 2m = 12.45 and 3c + m = 19.70.
From the first equation, c = 12.45 − 2m. Substitute into the second: 3(12.45 − 2m) + m = 19.70, so 37.35 − 6m + m = 19.70, giving 37.35 − 5m = 19.70. Then −5m = −17.65, so m = 3.53. The correct answer is A.
Which system of linear equations has no solution?
In option A, multiplying the first equation by −1 gives 5x − 9y = 8, identical to the second equation — infinitely many solutions. In option C, the coefficient ratios (−15/5 = −3 and 27/9 = 3) don't match each other, so the lines intersect at one point.
In option D, multiplying the first equation by −2 gives 10x − 18y = −16, which matches the second equation's coefficients but not its constant (−16 vs. 16) — parallel lines, no solution. The correct answer is D.
9x + 5y = 19 and −9x − 3y = 7. The solution to the given system of equations is (x, y). What is the value of 2y?
Add the two equations directly — the x-terms cancel: (9x + 5y) + (−9x − 3y) = 19 + 7.
That gives 2y = 26 immediately.
31(x − n) = 31y + 31n. One of the equations in a system of two linear equations is given, where n is a positive constant. The system has no solution. Which equation could be the second equation in this system?
Simplify the given equation by dividing by 31: x − n = y + n, which rearranges to x − y = 2n, or (multiplying by 3) 3x − 3y = 6n.
For no solution, the second equation needs the same coefficient ratio (3 to −3) but a different constant than 6n. Option D, 3x − 3y = 3n, has that same ratio with constant 3n — different from 6n since n is a positive constant — making the lines parallel. (Option A is identical to the derived equation, giving infinite solutions instead.) The correct answer is D.
At what point (x, y) do the graphs of the equations y = 2x + 4 and y = 3x − 6 intersect in the xy-plane?
Set the two expressions for y equal to each other: 2x + 4 = 3x − 6.
Subtract 2x from both sides: 4 = x − 6, so x = 10. Substitute back: y = 2(10) + 4 = 24. The intersection point is (10, 24). The correct answer is C.
2x + 5y = 3 and 8x + 20y = 12. 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?
Notice the second equation is exactly 4 times the first (4 × (2x+5y) = 8x+20y, and 4 × 3 = 12), so both equations describe the same line, with infinitely many solutions.
Let y = r and solve the first equation for x: 2x = 3 − 5r, so x = 3/2 − 5r/2 = −5r/2 + 3/2. That gives the point (−5r/2 + 3/2, r). The correct answer is D.
3x − 2y = 5 and x = 7. What is the solution (x, y) to the given system of equations?
Substitute x = 7 into the first equation: 3(7) − 2y = 5, so 21 − 2y = 5.
Subtract 21 from both sides: −2y = −16, so y = 8. The solution is (7, 8). The correct answer is A.
y = −(1/3)x and y = (1/5)x. The solution to the given system of equations is (x, y). What is the value of x?
Set the two expressions for y equal: −x/3 = x/5. Multiply both sides by 15 to clear denominators: −5x = 3x.
Add 5x to both sides: 0 = 8x, so x = 0. The correct answer is B.
y = −(1/7)x and y = (1/11)x. The solution to the given system of equations is (x, y). What is the value of x?
Set the two expressions for y equal: −x/7 = x/11. Multiply both sides by 77 to clear denominators: −11x = 7x.
Add 11x to both sides: 0 = 18x, so x = 0. The correct answer is B.
x = 2 + 8y and x = 4 − 8y. The solution to the given system of equations is (x, y). What is the value of 2x?
Since both equations equal x, set them equal to each other: 2 + 8y = 4 − 8y.
Add 8y to both sides: 2 + 16y = 4, so 16y = 2, giving y = 1/8. Substitute back: x = 2 + 8(1/8) = 2 + 1 = 3. Then 2x = 6.
3x + 5y = 21 and y = 3. What is the solution (x, y) to the given system of equations?
Substitute y = 3 into the first equation: 3x + 5(3) = 21, so 3x + 15 = 21.
Subtract 15 from both sides: 3x = 6, so x = 2. The solution is (2, 3). The correct answer is D.
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