Solving for a Variable in Terms of Others

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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:

1. One-step problems

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

$y$ is being added to $x$ .
Subtract $y$ from both sides:

$x = 8 - y$

2. Multiple-step problems

Example: Solve for $x$ in $2x - y = 12$

First, isolate the $x$ -term. Add $y$ to both sides: $2x = 12 + y$
Then divide both sides by 2: $x = \dfrac{12 + y}{2}$

3. Equations with multiple x-terms

Example: Solve for $x$ in $5x - 7 = 2xy + z$

Get all the $x$ -terms on one side: $5x - 2xy = z + 7$
Factor out $x$ : $x (5 - 2y) = z + 7$
Divide both sides by (5 – 2y): $x = \dfrac{z + 7}{5 - 2y}$

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.