Mathfolis

Linear Equations in One Variable

Unit 1 · Algebra

What the SAT asks here

A linear equation in one variable is any equation that reduces to ax + b = 0, with one unknown. On the Digital SAT, the Algebra domain spends the most time on this single skill — about 13–15% of the Math section. Calculator-allowed throughout, but every item in this topic is solvable by hand in under two minutes if you keep your distribution and sign work clean.

Standard form

One-variable linear equation
ax+b=cax + b = c
With variable on both sides
ax+b=cx+dax + b = cx + d

Solve by isolating x

Subtract the constant, then divide by the coefficient
ax+b=c  ⟹  x=c−baax + b = c \;\Longrightarrow\; x = \tfrac{c - b}{a}
Move the x-term, then isolate
ax+b=cx+d  ⟹  x=d−ba−cax + b = cx + d \;\Longrightarrow\; x = \tfrac{d - b}{a - c}
SAT Tip: If the SAT asks for the value of an expression (e.g., 3x + 5) rather than x itself, solve for x first, then substitute. Looking for shortcuts can backfire when the expression isn't an obvious multiple of the equation.
Caution: Distribute before combining like terms. The most common distractor on this topic is the answer you'd get from forgetting to distribute a negative sign across parentheses.
Type 1

Solve for the variable

Isolate x in a single linear equation. Move every x-term to one side, every constant to the other, then divide. Watch the sign of each term as you transfer it across the equals sign.

Example 1
If 4x − 7 = 2x + 9, what is the value of x? (A) 1 (B) 4 (C) 8 (D) 16

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Type 2

Value of an expression

The equation gives you enough information to determine x, but the question asks for the value of a related expression (like 3x + 5). Solve for x first, then substitute into the requested expression.

Example 2
If 5(x − 2) = 15, what is the value of 3x + 4? (A) 13 (B) 17 (C) 19 (D) 22

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Type 3

Real-world setup

Translate a short SAT-style scenario into a one-variable linear equation. Identify the unknown, write the equation, then solve. Re-read the question to make sure you're returning the quantity the prompt asked for, not just the variable.

Example 3
A taxi charges a fixed fare of 3.50plus3.50 plus2.25 per mile. Maya's total fare was $19.25. How many miles did Maya travel? (A) 6 (B) 7 (C) 8 (D) 9

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