Solve one-variable equations

SAT equations can bury a simple answer under fractions, parentheses, and several steps of algebra. When there's one unknown and the question asks for its value, Desmos can often find it without you rearranging anything.

Fast Desmos solution

Put each side on its own line in Desmos:

and

The graphs cross where $x=23$. At that $x$, the two sides are equal, so $23$ solves the equation, and the answer is C.

Here the answer was $x$ itself, but that isn't always so. The SAT might ask for the positive solution, the greater of two, or a value you build from the solution. So think of it as two jobs: solve, then answer the question.

A solution is a value of $x$ that makes the left side and the right side equal. So give each side its own graph. For an equation

graph

and

Where the two graphs meet, the sides have the same value. So the x-coordinate of that meeting point, the intersection, solves the equation.

Try it with

Enter

and

The graphs meet at $(12,47)$, so the solution is $x=12$.

Your equation has only one unknown, $x$. The $y$ is a helper Desmos uses to compare the two sides. (When $y$ is also an unknown in the original problem, you're solving a system of two equations.)

Replace other variable letters with x

Desmos uses lowercase $x$ as the variable on its horizontal axis. So if your equation uses a different letter, swap every copy of that letter for $x$.

Take

Swap $z$ for $x$:

Then graph y=3x+8 and y=29. They meet where $x=7$, so the original unknown is $z=7$.

Some equations have more than one real solution, and graphing both sides shows you all of them. Take

Enter

and

The equation is true wherever the two graphs have the same output, and that happens twice:

So the solutions are the $x$-coordinates:

Here both points fit on the screen, but that won't always happen. A second intersection can sit outside the window. So before you decide you've found every solution, zoom out or drag the graph around. Find them all first, and only then let words like positive, least, or greatest pick one.

Careful with the other coordinate, too. The $11$ is the value both sides share, not a solution, so don't submit a $y$-coordinate unless the question asks for it.

Desmos labels points with decimals. When the question needs an exact answer, like a fraction or a radical, and Desmos shows a decimal that doesn't end, don't guess the exact form from the screen. Convert it to a fraction if Desmos gives a clear one, check an answer choice, or finish with algebra.

Questions that ask whether an equation has no solution, one solution, or infinitely many need extra care. You'll get to them later, in How many solutions?

Direct entry can be a shortcut

You can also type the whole equation on one line. This is direct entry. Type

and Desmos draws vertical lines at $x=-6$ and $x=5$, the two solutions. That saves time when the lines are easy to click.

The two-graph method still gives the clearer picture, though. Stick with it when the equation might have several solutions, when the question asks you to compare them, when the shape of each side helps explain the answer, or when you want to see for yourself that the two sides are equal.

Whichever way you enter it, type the equation exactly as it's given. Don't cancel a variable factor or square both sides to make it easier to type. Changes like these can quietly remove a restriction or add solutions that don't work in the original equation.