SAT Infinitely Many Solutions: Find k, Check on Desmos

A system of two linear equations has infinitely many solutions when both equations describe the same line, so every coefficient in one is the same multiple of the other. That includes the constant term. These questions hide one or two constants, such as k, and ask which value makes the two lines match. You find the value with ratios and check it with a graph.

You can run each check below in the College Board version of Desmos on our Desmos on the SAT page. This guide gives you the rule, three worked examples and five practice problems with answers.

The rule: same line, same ratios

Write both equations in the form Ax + By = C. Then compare the matching parts: x-coefficient to x-coefficient, y-coefficient to y-coefficient, constant to constant.

In symbols, a₁x + b₁y = c₁ and a₂x + b₂y = c₂ have infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂. If one of the coefficients is 0, skip the division and ask a simpler question: what number turns the first equation into the second?

College Board allows a calculator on the entire Math section, and Bluebook has a Desmos calculator built in with a graphing option. There is no separate calculator section on the digital SAT, so you can graph any of these questions on screen.

Worked example 1 (I do): find two missing constants

SAT-style question. In the system ax + 4y = 12 and 6x + by = 18, a and b are constants. The system has infinitely many solutions. What are the values of a and b?

Step 1. Start with the pair you know. Both constants are given, 12 and 18. Their ratio is 12/18 = 2/3. The lines must be the same, so every first-to-second ratio has to equal 2/3.

Step 2. Solve for a. a/6 = 2/3, so a = 4.

Step 3. Solve for b. 4/b = 2/3, so 2b = 12 and b = 6.

Step 4. Check it on Desmos. Type each equation on its own line, exactly as the question writes it:

1  ax + 4y = 12
2  6x + by = 18

Desmos offers to add a slider for each letter. Add them, and each slider shows up as its own line, set to 1. Change those lines to a = 4 and b = 6. The two graphs now lie on top of each other as one line. Change a to 5 and the lines cross at a single point, which is one solution.

A new slider runs from -10 to 10. Typing the exact value on the slider line lands the constant where you want it, so you don't have to drag and hope.

By-hand check. Multiply 4x + 4y = 12 by 3/2 and you get 6x + 6y = 18, the second equation. Test a point too: (3, 0) gives 4(3) + 4(0) = 12 and 6(3) + 6(0) = 18. Both equations hold, so the answer is a = 4 and b = 6.

Worked example 2 (we do): the constant hides in a line through two points

SAT-style question. Line k is defined by 165x + py = w, where p and w are constants. A system made of line k and a second line has infinitely many solutions. The second line passes through (-9, 12) and (0, 1). What is the value of p + w?

Infinitely many solutions means line k is the second line. So you find the second line first, then match line k to it.

Step 1. Slope = (1 - 12)/(0 - (-9)) = -11/9. The point (0, 1) sits on the y-axis, so the y-intercept is 1. The second line is y = -(11/9)x + 1.

Step 2. Solve 165x + py = w for y: y = -(165/p)x + w/p.

Step 3. Match the slopes: 165/p = 11/9, so p = 165 × 9/11 = 135.

Step 4 (your step). Match the intercepts: w/p = 1. Find w, then add p + w.

Check your answer: w/135 = 1, so w = 135 and p + w = 135 + 135 = 270.

To check on Desmos, plot the two points and your line:

1  (-9, 12)
2  (0, 1)
3  165x + 135y = 135

The line on row 3 passes through both points. Mistype p as 136 and the line misses them, which tells you to recheck Step 3.

For more on this skill, work through the free Systems of Two Linear Equations lesson notes.

Worked example 3: flip the question to no solution

SAT-style question. The system 4x - 8y = 12 and ax - 2y = c has no solution. What must be true about a and c?

No solution means parallel lines. The x and y ratios match, and the constant ratio does not.

The y-coefficients set the scale: -8/-2 = 4. The x-coefficients need the same ratio, so 4/a = 4 and a = 1. If c were 3, the constant ratio would be 12/3 = 4 as well, and you would have one line drawn twice. So c cannot be 3.

Answer: a = 1 and c ≠ 3.

1  4x - 8y = 12
2  x - 2y = c

Add the slider for c. At c = 3 the lines merge into one. At c = 5, or any value other than 3, you see two parallel lines.

One more pattern is worth spotting on sight: y = (k/3)x + 5 and 3y = kx + 15. Divide the second equation by 3 and you get the first. They are the same line for any k, so the system has infinitely many solutions whatever k is.

Your turn (you do): 5 practice problems

1. The system 3x + ky = 15 and 6x + 4y = c has infinitely many solutions. Find k and c.
2. The system 5x - 10y = 20 and mx - 2y = d has no solution. Find m, and give the one value d cannot equal.
3. The system px + 3y = 12 and 4x + qy = r has infinitely many solutions, and its line passes through (3, 0). Find p + q + r.
4. In the system 2x - 7y = 5 and kx - 21y = 15, k is a constant. For what value of k does the system have infinitely many solutions?
5. How many solutions does the system 6x - 9y = 12 and 4x - 6y = 10 have?

Answers

1. k = 2, c = 30. The x-coefficients show the second equation is 2 times the first, so 2k = 4 and c = 2(15).
2. m = 1, and d cannot be 4. The y ratio is -10/-2 = 5, so 5/m = 5. With d = 4 the constants match too, and the system becomes one line.
3. 19. The point (3, 0) gives 3p = 12, so p = 4. Then 4x + 3y = 12 and 4x + qy = r must be the same equation, so q = 3 and r = 12, and 4 + 3 + 12 = 19.
4. k = 6. The y ratio is -21/-7 = 3 and the constant ratio is 15/5 = 3, so k = 3 × 2 = 6.
5. No solution. The x and y ratios are both 3/2, but the constant ratio is 12/10 = 6/5. The lines are parallel.

When algebra is faster

College Board's calculator guidance says some Math questions are better done without a calculator, even though one is allowed. We put most "find k" ratio questions in that group: one division gives you the multiple, and one more line gives you k. Use Desmos to confirm the answer, or when you can't see the multiple at a glance.

Mistakes to avoid

• Matching only the slopes. Equal x and y ratios give parallel lines, which means no solution. For infinitely many solutions the constant ratio has to match as well. If Desmos shows two parallel lines, you stopped one step early.
• Dropping a minus sign. 2x + 4y = 8 and 2x - 4y = 8 share an x-coefficient, but their slopes are -1/2 and 1/2, so they cross once. Type each minus sign exactly as the question shows it.
• Answering the wrong quantity. In worked example 2, p = 135 is not the answer. The question asks for p + w, which is 270.

Practice next

Start with the free Systems of Two Linear Equations lesson notes. Then try the free Solution Counts and Matching Terms lesson notes, which cover solution counts for more kinds of equations, including absolute value equations. For more questions, browse the SAT question bank.

Related guides: if your system has one solution, see how to solve a system of equations on Desmos. For shaded regions, read SAT inequalities on Desmos. For equations with powers, see how to count solutions of polynomial equations and the sum of solutions formula.

Checked on September 25, 2026 against College Board's SAT Suite calculator policy, College Board's March 12, 2024 digital SAT launch release, the College Board version of the Desmos graphing calculator and the Desmos Help Center article on sliders. Every answer on this page was recomputed before publishing. The Test Advantage is independent of College Board and Desmos.