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Equations With
One Variable

Master the art of isolating the unknown and solving equations that appear throughout the SAT Math section.

SAT Math

What Is an Equation?

An equation is a mathematical statement that two expressions are equal. When an equation contains a single unknown (like \( x \)), your job is to find the value of that variable that makes the statement true.

\( 3x + 5 \)
\( = \)
\( 20 \)
 Left Side Right Side

Key Vocabulary

Variable
A letter (like \( x \)) representing an unknown value you need to find.
Coefficient
The number multiplied by the variable. In \( 7x \), the coefficient is \( 7 \).
Constant
A fixed number with no variable attached, like \( 12 \) or \(-3\).

Why This Matters for the SAT

One-variable equations are the most frequently tested algebra skill on the SAT. They appear as standalone problems and as sub-steps inside word problems, geometry questions, and data analysis.

Linear
\( ax + b = c \): one step or multi-step, fractions, decimals
Rational
Variables in denominators: \( \frac{k}{x} = m \)
Absolute Value
\( |ax + b| = c \): splits into two cases

The Golden Rule of Equations

\( 3x + 5 \)
\( 20 \)
 Same operation on both sides

Inverse Operations

To isolate \( x \), apply the inverse (opposite) operation to undo each step. Work from the outermost operation inward toward \( x \).

Operation Inverse Example
Addition (\( +b \)) Subtraction (\( -b \)) \( x + 7 = 12 \) → subtract \( 7 \)
Subtraction (\( -b \)) Addition (\( +b \)) \( x - 3 = 10 \) → add \( 3 \)
Multiplication (\( \times a \)) Division (\( \div a \)) \( 5x = 40 \) → divide by \( 5 \)
Division (\( \div a \)) Multiplication (\( \times a \)) \( \frac{x}{4} = 9 \) → multiply by \( 4 \)

Linear vs. Non-Linear

Linear Equation
Variable has exponent \( 1 \) (no \( x^2 \), no \( \sqrt{x} \)). The graph is a straight line. Example: \( 2x + 3 = 11 \)
Non-Linear Equation
Variable has an exponent other than \( 1 \) or appears in a denominator, radical, or absolute value. Example: \( x^2 = 25 \)

Step-by-Step Strategy

1
Simplify each side. Distribute parentheses and combine like terms.
2
Collect variable terms on one side and constants on the other using inverse operations.
3
Isolate the variable by dividing (or multiplying) both sides by the coefficient.
4
Check your answer by substituting it back into the original equation.

Example 1: Basic Linear

1
Subtract 7 from both sides: \( 3x + 7 - 7 = 22 - 7 \) → \( 3x = 15 \)
2
Divide by 3: \( \frac{3x}{3} = \frac{15}{3} \) → \( x = 5 \)
3
Check: \( 3(5) + 7 = 15 + 7 = 22 \) ✓

Example 2: Multi-Step

1
Distribute: \( 5x - 10 + 3 = 2x + 11 \) → \( 5x - 7 = 2x + 11 \)
2
Subtract \( 2x \): \( 3x - 7 = 11 \)
3
Add 7: \( 3x = 18 \)
4
Divide by 3: \( x = 6 \)

Example 3: Fractions

1
Find the LCD of 3 and 4: LCD = 12
2
Multiply every term by 12: \( 12 \cdot \frac{x}{3} + 12 \cdot \frac{x}{4} = 12 \cdot 7 \) → \( 4x + 3x = 84 \)
3
Combine: \( 7x = 84 \) → \( x = 12 \)

Visualizing the Solution

Every one-variable equation pinpoints a single location on the number line (unless it has no solution or infinitely many solutions).

0 3 6 9 12
\( x = 12 \)
Solution to \( \frac{x}{3} + \frac{x}{4} = 7 \)

Special Cases

Identity
\( 2(x+1) = 2x + 2 \) simplifies to \( 0 = 0 \). Infinitely many solutions: every real number works.
Contradiction
\( x + 3 = x + 5 \) simplifies to \( 3 = 5 \). No solution: no value of \( x \) works.
Conditional
\( 2x + 1 = 9 \) has exactly one solution: \( x = 4 \). This is the most common type on the SAT.

Common Mistakes

✕ Forgetting to Distribute
Writing \( 2(x+3) = 2x+3 \) instead of \( 2x+6 \). Always multiply by every term inside the parentheses.
✕ Sign Errors
Losing the negative when subtracting: \( 5 - (-3) = 2 \) is wrong. It equals \( 8 \). Double-check signs.
✕ Dividing Only One Side
Writing \( 4x = 20 \) → \( x = 20 \). You must divide both sides by \( 4 \) to get \( x = 5 \).

Summary Table

Equation Type Strategy Key Move
\( ax + b = c \) Two-step isolation Subtract \( b \), divide by \( a \)
\( ax + b = cx + d \) Collect variables Move \( x \)-terms to one side
With parentheses Distribute first Apply distributive property
With fractions Clear denominators Multiply by LCD
\( |ax+b| = c \) Split into two cases Positive case and negative case

Question 1 of 5

Solve for \( x \): \( 5x - 8 = 17 \)

Question 2 of 5

Solve for \( x \): \( 3(x + 4) = 2x + 20 \)

Question 3 of 5

Solve for \( x \): \( \frac{2x}{5} - 3 = 7 \)

Question 4 of 5

How many solutions does the equation \( 4(x - 1) = 4x + 3 \) have?

Question 5 of 5

A plumber charges a flat fee of $40 plus $25 per hour. If the total charge is $165, how many hours did the plumber work?