Linear Equations in One Variable Pattern - Evaluate Expression

Finding the value of a related expression using a shortcut instead of solving for the variable first

The SAT loves to give you an equation and then ask for the value of a different expression — not the variable itself. The fast approach is to manipulate the equation so the target expression appears directly, skipping the step of solving for the variable. On harder questions the "expression" might be a nonlinear chunk like $k^2 + 3$ or a fraction like $\dfrac{p}{3} - 1$, and the equation is designed so you can treat that entire chunk as a single unknown.

 

Scale the Entire Equation

The simplest version: the given equation and the target expression differ only by a constant multiple. Multiply or divide both sides to jump straight to the answer.

$$18k - 6 = 24$$ What is the value of $3k - 1$?

Notice that $18k - 6 = 6(3k - 1)$. Divide the entire equation by 6:

$3k - 1 = \dfrac{24}{6}$, so $3k - 1 = 4$.

This also works when you need to scale up. If the target expression has a larger coefficient, multiply instead of dividing:

$$3y - 5 = 7$$ What is the value of $9y - 15$?

The target expression $9y - 15$ is $3(3y - 5)$. Multiply both sides by 3:

$9y - 15 = 3 \times 7 = $ $21$.

 

Isolate a Sub-Expression

Sometimes the target expression shares terms with the equation but isn't a simple multiple. Isolate the common piece, then build the target from there.

$$\frac{z}{2} + 6 = 10$$ What is the value of $\dfrac{z}{2} - 6$?

Both expressions contain $\dfrac{z}{2}$. Subtract 6 from both sides of the equation: $\dfrac{z}{2} = 4$.

Substitute into the target: $4 - 6 = $ $-2$.

When the equation involves an ax + b form and the target has the same ax but a different constant, isolate ax first:

$$6m + 1 = 25$$ What is the value of $6m - 5$?

Subtract 1 from both sides: $6m = 24$.

Substitute into the target: $24 - 5 = $ $19$.

 

Factor and Divide

When the equation's left side is a multiple of the target expression, factor first, then divide:

$$5y + 20 = 50$$ What is the value of $y + 4$?

Factor the left side: $5(y + 4) = 50$. Divide both sides by 5:

$y + 4 = \dfrac{50}{5} = $ $10$.

 

Take the Reciprocal

If the equation gives you a fraction and asks for its reciprocal, just flip both sides:

$$\frac{z + 1}{5} = 2$$ What is the value of $\dfrac{5}{z + 1}$?

The target is the reciprocal of the left side. Flip both sides: $\dfrac{5}{z + 1} = \dfrac{1}{2}$, so the answer is $\dfrac{1}{2}$.

 

Treat the Expression as a Single Variable

On medium and hard questions, the same expression appears in multiple places. Substitute a single letter for the whole chunk, then solve the simpler equation.

$$8(y - 7) = 7(y - 7) + 15$$ What is the value of $y - 7$?

Let $u = y - 7$. The equation becomes $8u = 7u + 15$.

Subtract $7u$ from both sides: $u = 15$, so $y - 7 = 15$.

This works even when the inner expression isn't linear:

$$6(t^2 + 3) = 5(t^2 + 3) + 16$$ What is the value of $t^2 + 3$?

Let $u = t^2 + 3$. The equation becomes $6u = 5u + 16$.

Subtract $5u$: $u = 16$, so $t^2 + 3 = 16$.

The same idea applies when the expression is nested inside fractions:

$$-3\left(1 - \frac{w}{2}\right) + 8 = -5\left(1 - \frac{w}{2}\right) + 18$$ What is the value of $1 - \dfrac{w}{2}$?

Let $u = 1 - \dfrac{w}{2}$. The equation becomes $-3u + 8 = -5u + 18$.

Add $5u$ to both sides: $2u + 8 = 18$. Subtract 8: $2u = 10$, so $u = 5$.

Therefore $1 - \dfrac{w}{2} = 5$.

 

The "Same Numerator, Different Denominator" Trick

The hardest variant puts the same expression over two different denominators and sets them equal. The only way $\dfrac{A}{B} = \dfrac{A}{C}$ with $B \neq C$ is if $A = 0$.

$$\frac{2k - 5}{7} = \frac{2k - 5}{4}$$ The value of $2k - 5$ is between which of the following?

The numerator $2k - 5$ is the same on both sides, but the denominators (7 and 4) are different. The only solution is $2k - 5 = 0$.

$2k - 5 = 0$, which is between $-1$ and $1$.

 

The SHORTCUT Method

1. Compare the equation to the target expression. Is the target a multiple, a factor, or does it share a sub-expression with the equation?
2. If it's a multiple or factor — multiply or divide both sides of the equation to produce the target directly.
3. If they share a sub-expression — isolate that sub-expression first (find $ax$ or $ax + b$), then plug it into the target.
4. If the same chunk appears multiple times — substitute a single letter for the chunk and solve the simpler equation.
5. If the same expression appears over two different denominators — the expression equals zero.

 

Watch Out For

• Giving the value of the variable instead of the expression. The question asks for $3k - 1$, not $k$. If you solve all the way to $k$, you still need one more step — plug it back in.
• Forgetting to scale both parts. When $18k - 6 = 24$ and you divide by 6, you must divide both terms on the left: $18k \div 6 = 3k$ and $-6 \div 6 = -1$. Missing one term gives a wrong answer.
• Distributing when you should factor. If you see $5y + 20 = 50$ and need $y + 4$, factoring to $5(y + 4) = 50$ is one step. Distributing and solving for $y$ works too, but it's slower and opens the door to arithmetic mistakes.
• Assuming the expression can't be zero. In $\frac{A}{7} = \frac{A}{4}$, students often cross-multiply and get lost. The direct insight — $A$ must be zero — is faster and avoids errors.