Desmos for the SAT · Technique 1 of 23

Solve Any Equation by Graphing

Turn "solve for x" into "find where two graphs cross".

SolvesSolve for xLinear equationsNonlinear equationsExponential equationsWhere two graphs cross

Watch the lesson

The whole move worked on one real SAT question, narrated.

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The same question, narrated by Nova: both sides on their own lines, then click where they cross.

Here's the single move that saves the most time on SAT math: stop solving equations, and start using Desmos to find where two graphs cross.

In Desmos, put the left side of the equation on one line, the right side on another, and read the x-value where they meet. That x is your answer.

The move works the same whether the equation is a clean line, something messier with an x-squared, or even an exponential like $2^{x} = 32$ . Two lines every time, and you skip almost every step where a sign error could creep in.

Desmos graphs in x and y, so if the problem uses a different letter like a or k, swap it for x. Keep the two sides on two lines. A one-variable equation typed on a single line does graph, as vertical lines, but Desmos marks no clickable point on it, so the two-line crossing is the route that hands you a number.

Nova

If you learn one Desmos move, make it this one. It shows up all over the Math section, and it turns "solve for x" into "click the dot."

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps rearranging to zero and factoring.

The question

$x^{2} + 6x = 16$

What is the positive solution?

The algebra way

Move everything to one side: x^2 + 6x - 16 = 0.
Find two numbers multiplying to -16 and adding to 6, which is the step that either comes to you or does not.
Factor: (x + 8)(x - 2) = 0, so x = -8 or x = 2.
Pick the one the question asked for. Both roots are on the page as answer choices.

The Desmos way

Type y = x^2 + 6x and y = 16 as two lines.
Click each crossing: they sit at $x = -8$ and $x = 2$ .
The question wants the positive one, so 2.

The payoff: Nothing is rearranged and nothing is factored, so the step that needs a flash of insight disappears. Both solutions are on screen at once, which also makes it obvious there are two.

Step by step

The whole move, all 4 steps

Put one side of the equation on its own line as y = ..., exactly as it is written.
Put the other side on the next line, also as y = .... There is never anything to rearrange first.
Click inside either equation box to reveal the gray dots, then click the one where the two graphs cross.
Read the last line of the question and answer what it actually asks for, which is often not plain x.

The question

The equation $x^{2}+6x=16$ is given.

What is the positive solution to the equation?

Step 1 of 5

Type the left side on its own line as y = x² + 6x, exactly as written. Nothing moves to the other side and nothing is set equal to zero.

In the calculator: y=x^2+6x

The equation is not in the form algebra class wants. Desmos does not care.

Now try it on a real SAT question

Head or scratch paper. Faster in your head, or on scratch paper, than the calculator takes to open. Add 8, divide by 4, done. Knowing when not to reach for Desmos is worth as many points as knowing how to use it.

The equation 4y - 8 = 12 can be solved for y.

What is the value of y?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

If a question asks for a constant that makes a system have no solution or infinitely many solutions, that is the systems lesson, not this one, and sliders are finicky and estimation-prone. Set the slopes or coefficients equal algebraically instead.
If the equation factors in a couple of seconds, factoring is faster than graphing and clicking.

All Desmos techniques

Put this move into real practice

"Solve Any Equation by Graphing" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.