How do you solve a nonlinear system on the SAT with Desmos?

Type both equations into the Desmos graphing calculator, one per row, for example y = x^2 − 4 and y = x − 2. Click either graph and Desmos marks the points where the parabola and the line cross. Each crossing is a solution to the nonlinear system: here (−1, −3) and (2, 0).

How do you find the y-value of a solution to a nonlinear system in Desmos?

Click the crossing that matches the condition the question gives, such as the solution with a negative x-coordinate, and read its second coordinate. For y = x² − 4 and y = x − 2, the solution with x < 0 is (−1, −3), so the y-value is −3.

How do you find k so a line and a parabola have exactly one solution using Desmos?

Type the line with the letter left in, such as y = 2x + k, and accept the slider Desmos offers for k. Drag it until the line touches the parabola at a single point. For y = x² − 4, that happens at k = −5, with the line touching at (1, −3).

Why does Desmos only show one solution to my SAT nonlinear system?

The second crossing is often off screen, because parabolas climb quickly out of the default window. A slanted line that cuts into a parabola has to cross it twice unless it only touches it, so zoom out before you decide the system has one solution.

How to Solve SAT Nonlinear Systems with Desmos

Q: How do you find k so a line and a parabola have exactly one solution using Desmos?

Type the line with the letter left in, such as y = 2x + k, and accept the slider Desmos offers for k. Drag it until the line touches the parabola at a single point. For y = x² − 4, that happens at k = −5, with the line touching at (1, −3).

Q: Why does Desmos only show one solution to my SAT nonlinear system?

The second crossing is often off screen, because parabolas climb quickly out of the default window. A slanted line that cuts into a parabola has to cross it twice unless it only touches it, so zoom out before you decide the system has one solution.

Last updated: October 7, 2026

Type both equations into Desmos, one per row, and click where the parabola and the line cross. Each crossing is a solution to the system, and Desmos labels it with both coordinates. A line can hit a parabola twice, once or not at all, and the screen shows you which before you do any algebra.

Practice Nonlinear Equations and Systems now →

The problem

Solve: x² − 4 = x − 2
Solution: x = −1 and x = 2

On the SAT this usually arrives as a system: y = x² − 4 and y = x − 2. The two graphs cross at (−1, −3) and (2, 0).

What do you type?

Two rows, copied as the question prints them:

Row 1: y = x^2 − 4
Row 2: y = x − 2

Desmos draws a parabola opening upward and a line cutting through it. Click either graph and gray dots appear where they meet. Click each dot and it stays labelled: (−1, −3) and (2, 0).

If the line comes as x − y = 2, type that. Desmos graphs it as it stands, and solving for y first is time you don't get back.

Which number does the question want?

A one-variable quadratic only ever wants x. A system can ask for any of these:

"What are the solutions?" Both points: (−1, −3) and (2, 0)
"If (a, b) is a solution and a < 0, what is b?" Find the crossing with negative x, then give its y: −3
"What is the sum of the y-coordinates of the solutions?" −3 + 0 = −3
"What is one possible value of x?" Either −1 or 2

The "(a, b)" kind is the one that costs points. It names a condition on one coordinate and asks for the other. Find the dot that matches the condition first, then read the coordinate you were asked for.

Practice Nonlinear Equations and Systems now →

What if a crossing is off the screen?

The default window is small, and parabolas climb fast, so a second crossing can sit up at y = 60 where you can't see it.

If you only see one crossing, zoom out before you commit. A slanted line that cuts into a parabola has to come back out the other side unless it only touches it, so one visible crossing usually means there's another one off screen.

How do you handle "exactly one solution"?

The SAT likes to hide a constant in the line: "y = x² − 4 and y = 2x + k have exactly one solution. What is k?"

Type y = 2x + k. Desmos offers a slider for k. Click it, then drag:

Two crossings: the line cuts through the parabola
One crossing: the line just touches it
No crossings: the line passes below without touching

Drag until the line touches the parabola at a single point. Here that happens at k = −5, touching at (1, −3). If the slider steps past the value, click the number beside k and type each answer choice in turn until one leaves a single gray dot.

The algebra behind that, substitution and the discriminant, is in SAT Nonlinear Equations and Systems. For a quadratic with no second equation, How to Solve SAT Quadratic Equations with Desmos covers reading the x-axis crossings.

What if the crossing isn't a whole number?

Desmos usually labels it as a decimal. Swap the line for y = x and the right-hand crossing reads (2.562, 2.562), while the choices say (1 + √17)/2. When the answer choices use roots, type each choice on its own row and Desmos prints its decimal value beside it. Match the decimal to the crossing. Don't try to work out which root it is from the decimal yourself.

How do you practice this?

Take 10 nonlinear system questions from practice tests, including at least two with a constant to find. For each one:

Type both equations as printed, then zoom out once before you count crossings
Click every crossing and write down both coordinates, not just x
Reread the question and circle which coordinate, and which crossing, it asked for

When you miss one, check whether you missed a crossing off screen or read the right dot for the wrong coordinate. Those two mistakes need different fixes.

HIROSCORE tracks nonlinear equations and systems as their own skill, separate from the rest of Advanced Math, so a habit of answering with x when the question wanted y shows up in your results instead of hiding inside a decent math score. The GPS for your SAT score.

Ready to practise Nonlinear Equations and Systems?

Practice Nonlinear Equations and Systems now →

Practice other SAT skills →Take the HIROSCORE diagnostic →

References

Digital SAT Suite Specifications Overview (College Board)
SAT Student Guide (College Board)
Desmos Help Center: Getting Started with the Desmos Graphing Calculator