Desmos intersection SAT method for solving equations by graphing both sides

By the Cheetah Prep team · Reviewed July 13, 2026

To solve an equation fast on the digital SAT, graph both sides in Desmos. The solution is the intersection, the $x$ value where the two graphs meet.

This works because an equation like $f(x) = g(x)$ is true only at inputs where the outputs match. In Desmos, type $y = f(x)$ and $y = g(x)$ on separate lines. Tap the intersection point and read the solution. This helps when the algebra is messy, when the equation is nonlinear, or when the solution is buried inside a weird expression.

Use this method when you see:

An equation that is hard to factor or simplify cleanly
Mixed functions, like a quadratic equal to a line, or a rational expression equal to a number
A question asking for solutions, roots, or where two expressions are equal

Quick accuracy tips:

Graph both sides as full functions, not one combined expression you might simplify wrong
Zoom until the intersection is easy to see, then tap the point to get precise coordinates
If you see more than one intersection, read the question carefully, it may want all solutions

If you want more Desmos calculator moves that rely on intersections and reading key points, start with the SAT Desmos guides.

When to use this Desmos method

Use the Desmos intersection method when the SAT gives you an equation in one variable, the algebra will take multiple risky steps, and the graphs will meet at a clear $x$ value.

Look for question patterns like these:

Two different expressions set equal: something like “solve $f(x) = g(x)$ ” where each side has its own structure, for example a square root on one side and a polynomial on the other. Graph $y = f(x)$ and $y = g(x)$ , then read the intersection.
Nonlinear meets linear: a line equal to a quadratic, exponential, or rational expression. These can force messy rearranging, but Desmos turns the solution into a point.
Nested or cluttered expressions: parentheses inside parentheses, fractions, and mixed operations. Graphing both sides cuts down mistakes from distributing, combining like terms, or clearing denominators incorrectly.
A solution that is not “nice”: if the answer choices are decimals or weird values, graphing is often faster than forcing exact algebra.

It is also a strong fit when the question asks for:

Where two expressions are equal
The $x$ value that makes an equation true
All solutions, if the graph shows more than one intersection

Skip this method if the problem is already one clean step (like a simple linear equation) or if the SAT clearly wants an exact form that a graph might only show approximately. For more practice spotting when the calculator is worth it, use free SAT practice.

Step by step in Desmos

Rewrite the equation as two sides
Look at the original equation and identify what is on the left side and what is on the right side. Do not start by moving everything to one side unless the problem is already in that form. The goal is to keep each side as its own expression so you can graph both sides directly.
Graph the left side as a function of x
In Desmos, type the left expression as an equation with y. Use parentheses carefully so the calculator follows the same order of operations as the problem.
y = (left side expression)
Graph the right side on a new line
On the next line, type y equals the right expression. Now the original equation is true exactly where these two graphs cross.
y = (right side expression)
Find the intersection point
Tap the point where the two graphs meet. Desmos will show the coordinates. The x coordinate is a solution to the original equation because it makes both sides equal.
Check for more than one solution
If the graphs cross in more than one place, each intersection gives a solution. Pan left and right and zoom out a bit to make sure you are not missing another crossing.
Make the window show the crossing clearly
If you do not see an intersection right away, use zoom and pan until both curves are visible in the same view. If the graph looks almost touching, zoom in so the intersection point is easier to tap and read.
Match what the SAT is asking for
Some questions want the x value, some want the y value, and some want both. Read the prompt carefully, then use the coordinate you tapped. If the answer choices are in a different form, use the same x value to evaluate one side and get the requested value.
Do a quick validity check
Plug the x value back into both expressions in Desmos to confirm they match. You can do this by typing each side with the x value substituted, or by evaluating each expression at that x value on a new line. This catches mistakes like a missed parenthesis or a mistyped exponent.

Exact expressions to enter

$y=f(x)$ Type this into Desmos
If the problem gives named functions, graph each side exactly as written.
$y=g(x)$ Type this into Desmos
Use a separate line for the other side so you can tap the intersection.
$y=\sqrt{x+3}$ Type this into Desmos
Example left side with a square root. Keep the entire radicand inside the root.
$y=2x-1$ Type this into Desmos
Example right side that is linear. The solution is the x coordinate of the intersection.
$y=\frac{x+1}{x-2}$ Type this into Desmos
Example rational expression. Watch for x values that make the denominator 0, those are not allowed.
$y=3$ Type this into Desmos
If one side is a constant, graph it as a horizontal line.
$y=\left(x-4\right)^2+1$ Type this into Desmos
Example quadratic in vertex form. Parentheses matter, type them.
$y=\frac{1}{2}x+5$ Type this into Desmos
Use 1/2 for a fraction coefficient, do not type 0.5 unless the problem uses decimals.

Worked SAT style example

Example

Solve for $x$ : $\sqrt{x + 5} = x - 1$ .

Graph both sides as two functions in Desmos.
On one line, enter $y = \sqrt{x + 5}$ .
On a second line, enter $y = x - 1$ .
Tap the intersection point of the two graphs.
Read the $x$ coordinate of the intersection. That $x$ value makes the two sides equal, so it solves the equation.
Quick check: substitute $x = 4$ into the original equation. Left side is $\sqrt{4 + 5} = 3$ , right side is $4 - 1 = 3$ , so it works.

Answer:

x = 4

Common mistakes

The intersection method is fast. Most wrong answers come from graphing the wrong thing or reading the graph wrong.

Forgetting the $y =$ part: If you type only an expression on a line, Desmos may treat it as a list or a constant. For solving, you want two full graphs: $y = f(x)$ and $y = g(x)$ .
Graphing a rearranged equation that changes the solutions: Turning $f(x) = g(x)$ into $f(x) - g(x) = 0$ is fine, but only if you do it perfectly. One algebra slip gives you a different graph. Graph both sides to avoid that risk.
Missing extra intersections: Nonlinear equations can have more than one solution. After you find one intersection, zoom out and look for other crossing points before you pick an answer.
Reading the wrong coordinate: The solution to the equation is usually the $x$ coordinate of the intersection, not the $y$ coordinate. If the question asks for the solution set, report the $x$ value or values.
Window problems, the intersection is off screen: If you do not see a crossing, it might still exist. Zoom out, pan, or adjust the window until both graphs show up over a wider range.
Domain issues from roots and denominators: Expressions like $\sqrt{x - 3}$ or $\frac{1}{x - 2}$ have restrictions. If you create an intersection in an invalid region, it is not a real solution.
Rounding too early: If Desmos shows $x \approx 2.67$ , keep the full value on screen while you check answer choices. Do not round until the final step.

When this method does not work

This method breaks down when the intersection is hard to see, off screen, or when the question needs an exact algebraic value instead of a graph based estimate.

Graphing both sides only helps if you can see the intersection clearly. On some SAT questions, Desmos will graph the expressions, but using intersections can waste time or push you toward the wrong answer.

Watch out for these situations:

No real solution: the graphs never meet. That can be the answer, but first make sure your window is not hiding the intersection.
Intersections that are hard to locate: if the graphs meet far left or right, you can lose time zooming and panning.
Too many intersections: periodic functions or wavy graphs can create many solutions. If the question wants all solutions in an interval, set the window to that interval and check every intersection inside it.
Near tangent intersections: if the graphs barely touch, the $x$ value can change with zoom and rounding. The point you click can shift.
Domain traps: square roots, denominators, and logs can restrict which $x$ values are allowed. Desmos will not flag extraneous solutions created by algebra steps. You still have to check that your intersection $x$ makes both original sides defined.
Exact form required: if the answer must be something like $3/4$ or $\sqrt{5}$ and the choices are in exact form, a decimal intersection can be hard to match.

When the intersection looks shaky, switch to algebra, or use targeted calculator techniques from the free SAT practice to get better at spotting these cases.

Practice questions

1.Use Desmos to solve by graphing both sides. What is the solution to $\sqrt{x+5}=x-1$ ?

2.Use Desmos intersection to solve. Which value of $x$ satisfies $(x+1)(x-3)=6$ ?

3.Solve by graphing both sides in Desmos. What is a solution to $\frac{10}{x}=x+1$ ?

4.Use Desmos to solve by graphing both sides. What is the solution to $x^2+2x+1=9$ ?

5.Graph both sides in Desmos. What is the $x$ value where the graphs of $y=2^x$ and $y=8$ intersect?

FAQ

What does “Desmos intersection SAT” mean?

It means you graph each side of an equation as its own function in Desmos. Then you find the point where the graphs intersect. The $x$ coordinate of that intersection is a solution because at that $x$ value, both sides give the same output.

How do I graph both sides of an equation in Desmos?

Rewrite the equation so each side is its own expression. Enter two lines: $y =$ (left side) and $y =$ (right side). Tap the intersection point. Read the $x$ value.

Why not just type the whole equation into Desmos?

If you type something like $f(x) = g(x)$ , Desmos treats it like a true false statement, not two graphs you can intersect. For intersection solving, you want $y = f(x)$ and $y = g(x)$ on separate lines so you can see where they match.

What if Desmos shows more than one intersection?

Then the equation has more than one solution. Tap each intersection and write down the $x$ values. If the question asks for “a solution,” use the answer choices or the context to pick the right one. If it asks for “all solutions,” you need every intersection that fits the domain shown or implied.

What if I do not see an intersection at first?

Zoom out. Pan to look for an intersection that is off screen. Check your parentheses. Make sure each side is graphed as a function of $x$ . If the graphs never meet, the equation has no real solution.

How accurate is the intersection method for SAT multiple choice?

It is accurate if you tap the intersection point and read the coordinate. Do not eyeball it. If the answer choices are close together, zoom in until the intersection is clear. Then tap again to get a more precise $x$ value.

Can I use this for nonlinear equations?

Yes. This method works especially well here. If the equation has squares, roots, rational expressions, or exponentials, graph each side and use the intersection to get the solution. That can save you from error prone algebra.

What common typing mistakes ruin the graph both sides equation method?

Missing parentheses and missing multiplication cause most of the problems. Type $(x+3)^2$ , not $x+3^2$ . Type $2(x+1)$ , not $2x+1$ if you meant to multiply. After graphing, do a quick reasonableness check: plug the intersection $x$ into both sides and confirm they match.

How can I confirm the $x$ value I read is really a solution?

Use substitution. Take the intersection $x$ value and plug it into both expressions. If both sides give the same number, it works. Do it with mental math for simple expressions, or use Desmos: put each expression on its own line and check the outputs match at that $x$ .

Does this work for systems of equations too?

Yes. A system is “graph both equations and find where they meet.” Enter both equations as $y =$ functions when you can, then use the intersection point. For systems, the full solution is the ordered pair $(x, y)$ , not the $x$ value.

About this page: written and reviewed by the Cheetah Prep team. Last reviewed July 13, 2026.