SAT Solving for a Variable in Terms of Others: Lesson, Practice Problems & Video Explanations

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By Glyn Caddell, BSME · Author of Digital SAT Math Secrets · Tutoring since 2008 — Updated August 18, 2026

Solving for a Variable in Terms of Others questions on the Digital SAT ask students to solve for a specific variable in an equation or formula in terms of the other variables. Sometimes the questions are relatively easy and only require one step, while complex ones require multiple steps, including factoring out a variable. Solving for a variable in terms of others questions appear on approximately 1.9% of the Digital SAT. Solving for a variable in terms of others falls under Advanced Math according to the College Board.

Table of Contents

Solving for a Variable in Terms of Others Lesson
Tips
Example
- Video Explanation
- Text Explanation
Practice Problems with Video Explanations
Common Mistakes
Related Topics

Solving for a Variable in Terms of Others Lesson

The lesson covers the types of Solving for a Variable in Terms of Others questions that show up on the Digital SAT. This lesson will go over how to approach these problems, how to solve one-step problems, how to solve multiple-step problems, and how to solve equations with multiple $x$ -terms.

What Do Solving for a Variable in Terms of Others Questions Look Like?

Here are some examples of Solving for a Variable in Terms of Others questions from the Digital SAT.

Which equation correctly expresses $x$ in terms of $b$ and $y$ ?
Which equation correctly expresses $k$ in terms of $j$ and $m$ ?

1. How to Approach These Problems

Think of the variable you’re solving for just like a number, and use inverse operations to isolate it. There are a few scenarios:

one-step equations
multiple-step equations
multiple $x$ terms

2. How to Solve One-Step Problems

Identify the variable you are trying to solve for, then consider what operation is being done to it.

For example, if a variable is being added to it, subtract that variable. If a variable is being multiplied by it, divide that variable.

Example: Solve for $x$ in $x + y = 8$

In the equation we want to solve for $x$ (isolate $x$ ).

$y$ is being added to the $x$ , so we have to subtract the $y$ to get $x$ alone.

$x + y = 8$

subtract $y$ from both sides, to get

$x = 8 - y$

This might seem confusing but consider the equation $x+4=8$ . If you had to solve for $x$ , you would simply subtract 4 from both sides.

Well, $y$ represents a number, so it should be treated the same way. The only difference is we can’t combine 8 and $y$ since they aren’t like terms, so it stays as $8-y$ .

3. How to Solve Multiple-Step Problems

Let’s look at a more complicated example.

Solve for $x$ in the equation $2x-y=12$ .

In this case, $x$ is being multiplied by 2 and $y$ is being subtracted from the $x$ -term.

We should solve for $x$ by first getting the $x$ -term by itself. That means add $y$ to both sides.

$2x - y = 12$

Add $y$ to both sides

$2x = 12+y$

Now, the $x$ -term, $2x$ , is by itself. When determining what to do next, completely ignore the right side of the equation and just focus on the $x$ -term.

We want to get $x$ by itself, so divide both sides of the equation by 2 to isolate the $x$ .

$2x = 12+y$

Divide both sides by 2 to get

$x = \dfrac{12+y}{2}$

If solving multi-step equations in terms of other variables is confusing, review solving algebraic equations first.

4. How to Solve Equations with Multiple x-terms

A problem that trips up a lot of students is when they have to solve for $x$ and there is an $x$ -term on both sides of the equation that can’t be combined.

The trick is to get all of the $x$ -terms alone on one side of the equation first and then factor out the $x$ .

Here is an example.

Solve for $x$ in the equation below.

$5x-7=2xy+z$

To solve this, let’s get all of the terms that have an $x$ onto the left side of the equation by subtracting $2xy$ on both sides.

$5x-7=2xy+z$

Subtract $2xy$ from both sides to get

$5x-7-2xy=z$

At this point all of terms with an $x$ are on the left side.

We want to get the $x$ -terms alone, so we should get rid of the $-7$ by adding 7 to both sides.

$5x-7-2xy=z$

Add 7 to both sides to get

$5x-2xy=z+7$

Now that we only have terms with an $x$ on the left side of the equation, we can factor out an $x$ .

$5x-2xy=z+7$

Factor out an $x$ from the expression on the left side to get

$x(5-2y)=z+7$

Since $x$ is being multiplied by the expression $(5-2y)$ , we should divide both sides by $(5-2y)$ to isolate the $x$ .

$x(5-2y)=z+7$

Divide both sides by $(5-2y)$ to get

$x=\dfrac{z+7}{5-2y}$

SAT Tips

Treat letters the same way you treat numbers. If it says “+y,” think of it like “+4.”
Be careful not to combine unlike terms (e.g., you can’t add 8 and y).
If multiple variables are attached to the one you’re solving for, factor it out.

Example

$5m=2n$

The given equation relates the positive numbers $m$ and $n$ . Which equation correctly expresses $m$ in terms of $n$ ?

A) $m=\dfrac{2n}{5}$

B) $m=\dfrac{5}{2n}$

C) $m=\dfrac{5n}{2}$

D) $m=10n$

Video Explanation

Text Explanation

Answer: A

To solve for $m$ , divide both sides of the equation by 5.

$5m=2n$ ,

$m=\dfrac{2n}{5}$

Common Mistakes

Two common mistakes students make when answering solving for a variable in terms of others questions are distributing the variable you are solving for into a set of parentheses and combining terms that have $x$ s but aren’t like terms.

Distributing the Variable You are Solving for Into a Set of Parentheses

A common mistake when solving for a variable in terms of others is distributing the variable you are solving for into a set of parentheses.

Distributing the variable into the parentheses makes solving for the variable more difficult.

For example, if you have to solve for $x$ in the given equation

$x(2t-3yz)=4$

Distributing $x$ into the parentheses would make the solving for $x$ more difficult.

The equation would become

$2tx-3xyz=4$

Instead, to solve for $x$ , you only need to divide the other expression on both sides.

$x(2t-3yz)=4$

Divide both sides by $2t-3yz$ to get

$x=\dfrac{4}{2t-3yz}$

Combining Terms That Both have xs But Aren't Like Terms

Another common mistake is combining terms that have $x$ s but aren’t like terms.

For example, an equation may have $3x+2xy$ . A mistake some students make is to combine the two terms and get $5xy$ or something similar. The two terms are not like terms, so they can’t be combined.