GMAT® Equations: How to Simplify and Solve Them

Most GMAT® quant questions involve an equation. Sometimes the test hands you one, sometimes you build it from a word problem, and sometimes it appears inside a Data Insights question. With that much riding on one skill, it makes sense that solving equations can look like a long list of moves to memorize: distribute here, move that term there, flip a sign somewhere in the middle.

The list is shorter than it looks. Nearly every equation on the exam yields to the same small set of rules, and those rules all grow out of one idea: the two sides of an equation are equal, and your job is to keep them that way. If algebra felt like symbol-shuffling back in school, you may not have been handed a system for seeing what each move preserves. That system is the rest of this guide, and it starts with what an equation is.

What an equation is

An equation is a statement that two quantities have the same value. $x = 5$ says it plainly, and so does $2x + 6 = 16$, just with more packed onto each side.

A few vocabulary pieces make the rest of this guide easier to follow:

• A variable is a symbol, usually a letter, that stands for a value that can change.
• A coefficient is a number multiplying a variable. In $3x$, the 3 is the coefficient of $x$.
• A term is one piece: a number, a variable, or a variable with its coefficient. In $3x + 5$, the terms are $3x$ and $5$.
• Like terms are terms built on the same variable. $3x$ and $5x$ are like terms. $3x$ and $5y$ are not.

Like terms matter because of one restriction: you can add or subtract only like terms. Three $x$'s plus five $x$'s makes eight $x$'s, the way three apples plus five apples makes eight apples. Three apples plus five oranges just makes a mixed pile; those terms stay separate. Multiplication plays by different rules, since $3x$ times $5y$ is $15xy$ whether or not the variables match. That last rule is a good flashcard: add and subtract only like terms, but multiply and divide any terms.

The golden rule of equations

Any operation you apply to one side of an equation, you apply to the other side too. It's sometimes called the golden rule of algebra, and an old-fashioned balance scale explains why. A scale with two pans holds equal weights, and the beam sits level. Add three pounds to the left pan, and you have to add three pounds to the right pan if you want the beam to stay level. Adding 3 to both sides of an equation works the same way: both sides look different afterward, but the relationship between them hasn't moved.

The scale also shows what solving is. Starting from $2x + 6 = 16$, you subtract 6 from both pans, then divide both pans by 2, and the beam stays level the whole way down to $x = 5$. Each valid move preserves the balance, and you keep making moves until the variable stands alone.

To make the idea stick, run it in reverse. Start with $x = 5$, add 3 to both sides, then multiply both sides by 2, then subtract a brand-new variable $z$ from both sides. The equation gets uglier at every step, and $x$ is still 5. That's what the intimidating equations on the test are: simple statements of equality with clutter layered on top, and none of your careful moves change the value hiding underneath. The moves don't create the value of $x$. They uncover it.

Two kinds of moves

Some moves need both sides, and some don't. The dividing line: does the move change the value of anything, or just rearrange it?

Moves that change values: adding 3, subtracting $2x$, multiplying by 5, dividing by 4. These need both sides, because each one tips the scale unless the other pan gets the same treatment.

Moves that just rearrange: distributing a factor into a sum, combining like terms within one side, rewriting a fraction with a common denominator. Rolling two one-pound lumps of clay into a two-pound ball doesn't upset a balanced scale, and these moves don't upset an equation. They can happen on one side alone.

Take distributing the 2 in $2(x + 3)$ to get $2x + 6$. The left side keeps its exact value in a new shape, so no balancing move is required. The confusion tends to show up mid-problem, when combining like terms on one side starts to look a lot like subtracting from both sides. The first rearranges what's already there. The second adds something new to both pans, and the second is the one that needs both sides.

Three rules that simplify any equation

When you're mid-problem and deciding what to do next, one question sorts it out: did this move change a value, or did it just rewrite one in a cleaner form? Value-changing moves obey the golden rule, do to one side what you do to the other. Rewriting moves, like distributing into a sum or combining like terms within one side, can happen on one side alone.

The golden rule needs a guardrail, though, because value-changing moves can still go wrong: add and subtract only like terms. If you add 3 to both sides of $x = 5$, the left side becomes $x + 3$, not $4x$. Combining the $x$ with the 3 you just added is the error that turns a true equation into a false one.

And the question of when to stop has its own answer: eliminate as many terms as you can. If no two terms on the page combine, and the variable stands alone, you're done. If there's still a like-term pair or a coefficient sitting on your variable, there's a move left.

The worked example below puts the three rules in order.

Worked example: variables on both sides

$2(x + 4) - 3 = 5(x - 2) + 7$

Distribute first, since multiplication hanging outside parentheses wants to happen right away. That's a rearranging move, one side at a time:

$2x + 8 - 3 = 5x - 10 + 7$

Combine the like terms on each side, which is the same clay-rolling move: $8 - 3$ on the left, $-10 + 7$ on the right.

$2x + 5 = 5x - 3$

Both sides now hold a variable term and a number term. Add 3 to both sides to gather the numbers on the left. That's a value-changing move, so both pans get it:

$2x + 8 = 5x$

Subtract $2x$ from both sides:

$8 = 3x$

Divide both sides by 3:

$x = \frac{8}{3}$

An answer that lands on a fraction is unremarkable on this exam, so a non-integer alone is no reason to re-run your work. One preference from a lot of years watching handwritten algebra: favor the order of moves that keeps negatives off the page, since they're error-prone when you compute by hand. Adding 3 before subtracting $2x$ above is a preference, not a rule; both orders reach the same equation.

Try this one

Add 4 to both sides, which gives $5x = 3x + 14$. Subtract $3x$ from both sides, which gives $2x = 14$, and dividing by 2 lands on $x = 7$, choice (C).

Choice (A) deserves a second look, because it's the answer you get by adding 4 to the right side only: $5x = 3x + 6$, then $x = 3$. Wrong answers on this exam are often built from one-sided operations, which is part of why the golden rule is the habit worth drilling. And the substitution check usually takes only a few seconds: $5(7) - 4 = 31$, and $3(7) + 10 = 31$, so both sides agree. Checking by substitution only works because an equation is a claim of equality, and a correct $x$ makes both sides land on the same number.

Where the test adds difficulty

The exam complicates equations in predictable ways, and the complications usually don't require new rules:

• Variables on both sides. Collect variable terms on one side and numbers on the other, using golden-rule moves, then divide out the coefficient. The worked example above is exactly this.
• Fractions in the equation. Multiply both sides by the common denominator to clear them. Multiplying changes values, so both sides need it, and after that the equation behaves like any other.
• Extra variables. Treat them like numbers you're carrying along. An equation like $2x + 3y = 12$ can be simplified but not solved to a single value, and some questions ask for the simplified relationship rather than a number.

Systems of equations and quadratic equations get their own lessons in the Math Basics series, since each is a skill of its own. Inequalities run on the same golden rule with one extra rule about when the sign flips, and our inequalities guide covers that piece.

A large share of equation questions also show up in the Data Insights section as Data Sufficiency, where the question is whether the given information pins the value down at all. Our Data Sufficiency guide walks that format.

FAQ

What is the golden rule of equations on the GMAT®?

Whatever operation you perform on one side of an equation, perform on the other side as well. Adding, subtracting, multiplying, or dividing changes the value of a side, so both sides need the change for the equality to survive.

Can you multiply terms that aren't like terms?

Yes. Multiplication and division work on any terms, like or unlike: $3x$ times $5y$ is $15xy$. The restriction applies to addition and subtraction, which combine only like terms.

How do you solve an equation with variables on both sides?

Use golden-rule moves to collect the variable terms on one side and the numbers on the other, then divide out the coefficient. From $2x + 5 = 5x - 3$, adding 3 and subtracting $2x$ from both sides gives $8 = 3x$, so $x = \frac{8}{3}$.

How do you handle fractions in a GMAT® equation?

Multiply both sides by the common denominator so the fractions clear. Because multiplying changes values, both sides need it. After that, the equation behaves like any other.

Should you always solve GMAT® equations with algebra?

No. When the answer choices contain variables, or a word problem's numbers would get messy, plugging in numbers or working back from the choices can be faster. Our guide on when picking numbers beats algebra covers the setups that signal it.

What's the best way to practice equation solving?

Official Guide questions, with every step written out. Our guide on using the Official Guide effectively covers how to work through them as a practice system. Writing the work down is what lets you find the exact step where an answer went sideways, which our guide on writing down more covers in detail. If the moves above feel rusty, the Math Basics series rebuilds them from zero.

Want to learn even more?

Related reading:

• GMAT® Inequalities: A Complete Guide
• GMAT® Algebra: What to Do When You're Stuck (Plugging In Numbers)
• GMAT® Data Sufficiency: How to Approach DS Questions
• How Writing Down More Improves Your GMAT® Score
• GMAT® Quant Preparation: How to Study for the Quant Section

The Math Basics series includes a full lesson on equations, "GMAT® Focus Edition Math Basics: Equations," and you can hear it on Spotify, Apple Podcasts, or YouTube.