How to Solve Systems of Equations on Desmos (SAT)

To solve a system of equations on Desmos, type each equation on its own line, click one of the graphs, and select the gray point where the graphs cross. That point's coordinates are the solution. Then answer what the question asks, which might be x alone, y alone, or something like r + s.

You can follow along on the College Board version of the calculator on our Desmos on the SAT page. Bluebook, the app you take the SAT in, has Desmos built in, and College Board allows a calculator on the whole Math section. Below, you watch one SAT-style system solved step by step, finish a second one with help, and then solve two on your own.

The three steps

1. Type each equation on its own line, as the question writes it. Desmos graphs an equation like 3x + 2y = 12 as typed, so you don't have to solve for y first.
2. Click a line or curve. Desmos puts gray dots on intercepts, intersections, maximums and minimums. Select the gray dot where the graphs cross to see its coordinates.
3. Answer the question that was asked. The point gives you x and y. If the question wants one coordinate, a sum, or a count of solutions, you finish that step.

One caution from Desmos's own help pages: coordinates show most reliably for equations written in y = form. If you click the lines and no gray dot appears where they cross, type each equation again as y = ... on a new line.

What the number of crossings tells you

• One crossing point: exactly one solution.
• No crossing: no solution. Two lines that never cross are parallel.
• Two crossing points: two solutions. You see this with a line and a parabola.
• One graph drawn on top of the other: infinitely many solutions. Our guide to systems with infinitely many solutions shows how to find the constant that makes this happen.

Example 1, worked for you: two lines

y = -10x + 13
y = -12x + 15

What is the solution (x, y) to the given system of equations?

SAT-style question from TTA's Digital SAT question bank (record 662977).

Line 1: y = -10x + 13
Line 2: y = -12x + 15

Both equations are already in y = form, so I type them as written. Every rearrangement is a chance to drop a minus sign, and here there is nothing to rearrange.

Before clicking, I predict. The slopes are -10 and -12. Different slopes mean the lines cross exactly once, so I am looking for one point.

I click either line, find the gray dot where the two lines meet, and select it. Its coordinates are (1, 3).

Then I check by hand: -10(1) + 13 = 3 and -12(1) + 15 = 3. Both equations give y = 3 when x = 1, so the solution is (1, 3). The question asks for the pair, so (1, 3) is the answer. If it had asked for y alone, the answer would be 3.

Example 2, you finish it: different letters and big numbers

20(0.6r - 0.8s) = -1,840
70(0.8r + 0.6s) = 6,580

The solution to the given system of equations is (r, s). What is the value of r + s?

SAT-style student-produced response question from TTA's Digital SAT question bank (record 663052).

Step 1: rename the letters. The Desmos graphing calculator uses x and y for the coordinate plane, and it reads r as the radius in polar coordinates. Type x in place of r and y in place of s. Write down that x means r and y means s.

Step 2: drop the commas inside numbers. Desmos uses commas to separate values, as in the point (1, 3). Type 1840, not 1,840.

Line 1: 20(0.6x - 0.8y) = -1840
Line 2: 70(0.8x + 0.6y) = 6580

Step 3: find the crossing. These lines cross at large values, so zoom out (the Zoom Out button, or set the axis bounds in Graph Settings) until you can see where they meet. If clicking a line shows no gray dot at the crossing, type the y = versions instead. You get them by dividing the first equation by 4 and the second by 14, then solving for y:

Line 3: y = (3x + 460)/4
Line 4: y = (470 - 4x)/3

Step 4 is yours. Select the crossing point, read its coordinates, and turn them into the value the question asks for. Remember what x and y stand for.

Check your answer: the lines cross at (20, 130), so r = 20 and s = 130. The question asks for r + s, so the answer is 150. By hand, the simplified equations are 3r - 4s = -460 and 4r + 3s = 470. Check them: 3(20) - 4(130) = 60 - 520 = -460, and 4(20) + 3(130) = 80 + 390 = 470. If you entered 20 or 130, you answered with one coordinate instead of the sum.

Your turn: two problems

Solve both before you read the answers below.

Problem A.
y = 2x² - 16x + 37
y + 4 = 0
How many solutions are there to the given system of equations?

SAT-style question from TTA's Digital SAT question bank (record 656780). In Desmos, type the exponent with ^, as in y = 2x^2 - 16x + 37.

Problem B.
3x + 2y = 7
x - 4y = 3
The solution to the system is (x, y). What is the value of x?

SAT-style practice problem written for this guide.

Answer to A: no solution (zero solutions). The second equation says y = -4, a horizontal line. Click the parabola and its lowest point shows as a gray dot at (4, 5). The parabola never goes below y = 5, so it never reaches the line y = -4. By hand: 2x² - 16x + 37 = -4 gives 2x² - 16x + 41 = 0. The discriminant is 16² - 4(2)(41) = 256 - 328 = -72, and a negative discriminant means no solution.

Answer to B: x = 17/7. The lines cross near (2.43, -0.14), and Desmos shows those coordinates as decimals. The SAT wants the exact value, so finish with substitution. The second equation gives x = 3 + 4y. Then 3(3 + 4y) + 2y = 7, so 9 + 14y = 7 and y = -1/7. That makes x = 3 + 4(-1/7) = 17/7. To confirm, type 17/7 on a new line in Desmos and compare its decimal, 2.428..., with the graph.

When algebra is faster

College Board's calculator tips say some Math questions are better done without a calculator. With systems, one clear case is when an equation already gives a variable, like y = -4 in problem A or x = 3 + 4y in problem B. Substitute and you are done in a line or two. Use Desmos to check the result. If a question asks for the sum of the solutions of one quadratic equation, a formula is faster than reading two crossing points; see our sum of solutions guide.

Mistakes to avoid

• Pasting a dash in place of a minus sign. Equations copied from a web page or PDF can carry a dash character that looks like a minus. Type the minus key yourself.
• Keeping the question's letters. Rename r and s, or a and b, to x and y, and drop commas inside numbers.
• Deciding "no solution" from the default view. A crossing can sit off screen. Zoom out, or check with algebra as in problem A.
• Answering with the whole point. Reread the last line of the question: it may want x, y, or a combination like r + s.

Practice next

Work through Algebra: Systems of Two Linear Equations, Notes and Practice, free lesson notes for this skill. For mixed Math practice, open the TTA SAT question bank. Related Desmos guides: find x- and y-intercepts on Desmos, SAT word problems on Desmos, and how to open the College Board version of Desmos.

Checked on September 25, 2026 against College Board's SAT Suite calculator policy and the Desmos Help Center pages Getting Started: Desmos Graphing Calculator, FAQs and Polar Graphing. We reworked every answer on this page by hand. The Test Advantage is independent of College Board and Desmos.