Linear Equations in One Variable Pattern - Solve for Variable

Solving for the variable in a linear equation

This pattern is the most fundamental skill on the SAT Math section: you're given an equation with one unknown and you need to isolate it. The difficulty ranges from simple one-step problems all the way to equations with fractional coefficients and common binomial factors. The core idea is always the same — undo what's been done to the variable by applying inverse operations in the right order.

 

One-Step and Two-Step Equations

The simplest version gives you an equation that's almost solved already. Just perform one inverse operation:

$$y + 15 = 165$$ What is the value of $y$?

Subtract 15 from both sides: $y = 165 - 15$, so $y = 150$.

When a coefficient sits in front of the variable, you need two steps — first isolate the variable term, then divide:

$$6x + 18 = 90$$ What value of $x$ is the solution to the equation?

Subtract 18 from both sides: $6x = 90 - 18 = 72$.

Divide both sides by 6: $x = \dfrac{72}{6}$, so $x = 12$.

Negative coefficients work the same way — just be careful with signs when you divide:

$$-10y + 120 = 70$$ Which of the following is the solution?

Subtract 120 from both sides: $-10y = 70 - 120 = -50$.

Divide both sides by $-10$: $y = \dfrac{-50}{-10}$, so $y = 5$.

 

Combine Like Terms First

Sometimes the equation has multiple variable terms on the same side. Combine them before you start isolating:

$$400 - 14m + 15m = 550$$ In the equation above, what is the value of $m$?

Combine the variable terms: $-14m + 15m = m$. The equation simplifies to $400 + m = 550$.

Subtract 400 from both sides: $m = 150$.

 

"Equivalent Equation" Questions

Instead of asking for the value of $x$, these ask which equation has the same solution. The trick is to recognize that simplifying the original equation produces an equivalent one. Divide both sides by the outside multiplier — don't distribute:

$$-6(5p - 2) = 90$$ Which equation is equivalent to the one shown?

Divide both sides by $-6$: $5p - 2 = \dfrac{90}{-6}$, so $5p - 2 = -15$.

That's the equivalent equation: $5p - 2 = -15$.

 

Common Binomial Factor with Fractions

When the same expression appears multiple times, factor it out instead of distributing. This is the key technique for medium-difficulty questions in this pattern:

$$\frac{1}{2}(x - 3) - \frac{1}{5}(x - 3) = 6$$ What is the value of $x$?

Factor out $(x - 3)$: $\left(\dfrac{1}{2} - \dfrac{1}{5}\right)(x - 3) = 6$.

Find a common denominator: $\dfrac{5}{10} - \dfrac{2}{10} = \dfrac{3}{10}$. So $\dfrac{3}{10}(x - 3) = 6$.

Multiply both sides by $\dfrac{10}{3}$: $x - 3 = 6 \cdot \dfrac{10}{3} = 20$.

Add 3: $x = 23$.

The same idea works when the variable is being subtracted inside the parentheses — just watch your signs at the end:

$$\frac{3}{4}(10 - z) - \frac{5}{6}(10 - z) = 1$$ What is the value of $z$?

Factor out $(10 - z)$: $\left(\dfrac{3}{4} - \dfrac{5}{6}\right)(10 - z) = 1$.

Common denominator of 12: $\dfrac{9}{12} - \dfrac{10}{12} = -\dfrac{1}{12}$. So $-\dfrac{1}{12}(10 - z) = 1$.

Multiply both sides by $-12$: $10 - z = -12$.

Subtract 10: $-z = -22$, so $z = 22$.

 

Common Binomial Factor with Integers

Harder versions use integer coefficients but add a twist — the inner expression may contain a coefficient on the variable, requiring an extra step after factoring:

$$9(2z - 1) - 3(2z - 1) = 42$$ What is the solution $z$ to the equation?

Factor out $(2z - 1)$: $(9 - 3)(2z - 1) = 42$, so $6(2z - 1) = 42$.

Divide both sides by 6: $2z - 1 = 7$.

Add 1: $2z = 8$. Divide by 2: $z = 4$.

 

Decimals with Distribution

The hardest variant gives you messy decimals on both sides. There's no shortcut — distribute, collect variable terms on one side, constants on the other, and divide:

$$1.2p - 0.88 = 5(p - 0.004) - 0.5$$ What is the value of $p$?

Distribute on the right: $1.2p - 0.88 = 5p - 0.02 - 0.5 = 5p - 0.52$.

Subtract $1.2p$ from both sides: $-0.88 = 3.8p - 0.52$.

Add 0.52 to both sides: $-0.36 = 3.8p$.

Divide: $p = \dfrac{-0.36}{3.8} \approx -0.0947$. As a fraction, $p = -\dfrac{9}{95}$.

 

The ISOLATE Method

1. Simplify each side — combine like terms, clear parentheses (distribute or factor), clear fractions if helpful.
2. Move variable terms to one side — add or subtract so all terms with the variable end up together.
3. Move constants to the other side — add or subtract to isolate the variable term.
4. Divide by the coefficient — whatever number is multiplied by the variable, divide both sides by it.

 

Watch Out For

• Sign errors when subtracting. In $-10y + 120 = 70$, you get $-10y = -50$, and dividing two negatives gives a positive answer. Students often lose track of the minus signs.
• Distributing when you should factor. If you see $\frac{1}{2}(x-3) - \frac{1}{5}(x-3) = 6$, factoring out $(x-3)$ is far faster than distributing both fractions. Look for repeated expressions.
• Stopping one step early. On factored problems, after finding $x - 3 = 20$, you still need to add 3 to get $x = 23$. Many wrong answers are the value of the inner expression, not the variable itself.
• Choosing a distractor operation. Easy questions are designed so that adding instead of subtracting, or multiplying instead of dividing, produces one of the wrong answer choices. Always check that your operation undoes what the equation does to the variable.