SAT Math Topics

Algebra › Linear Equations

Linear Equations

Easy

Concept Overview

A linear equation is an equation in which every variable appears to the first power. Solving one means finding the value that makes the equation true — we do this by isolating the variable using inverse operations.
The golden rule: whatever you do to one side, you must do to the other. Add, subtract, multiply, or divide — always both sides.
Special cases: if simplifying leads to a contradiction like 3=53 = 5 , there is no solution. If it leads to an identity like 0=00 = 0 , there are infinitely many solutions. On the SAT, you may be asked to find what value of a constant produces one of these cases.

Key Formulas

ax+b=c  ⟹  x=c−baax + b = c \implies x = \dfrac{c - b}{a}

Solve for x in a basic linear equation (a ≠ 0)

y=mx+by = mx + b

Slope-intercept form of a line (m = slope, b = y-intercept)

m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}

Slope between two points

Worked Example

Problem
If 3(x−2)=2(x+4)3(x - 2) = 2(x + 4) , what is the value of xx ?
Solution
  1. 1Distribute both sides: 3x−6=2x+83x - 6 = 2x + 8
  2. 2Subtract 2x2x from both sides: x−6=8x - 6 = 8
  3. 3Add 66 to both sides: x=14x = 14
Answer
x=14x = 14

Practice Problems

1

If 5x−3=2x+125x - 3 = 2x + 12 , what is the value of xx ?

2

Which of the following is the value of xx that satisfies 2x+43=6\dfrac{2x+4}{3} = 6 ?

3

For what value of kk does the equation kx+4=4x+kkx + 4 = 4x + k have infinitely many solutions?

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