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x 5

One- and Multi-Step
Equations

Master the art of isolating variables by applying inverse operations: from simple one-step problems to complex multi-step challenges.

SSAT Upper Level: Quantitative

What Is an Equation?

An equation is a mathematical statement that two expressions are equal. The equals sign \( = \) acts like a balance point: whatever is on the left side must have the same value as whatever is on the right side.

\(2x + 3\)
\(11\)
=

Key Vocabulary

Variable
A letter (like \( x \) or \( y \)) representing an unknown value we need to find.
Coefficient
The number multiplied by a variable. In \( 3x \), the coefficient is \( 3 \).
Constant
A number without a variable attached, such as \( 7 \) or \(-4\).

Why Equations Matter

Equations are the backbone of algebra and appear throughout the SSAT. Whether you are solving a word problem about ages, distances, or money, you will translate the situation into an equation and solve for the unknown.

Word Problems
Translate English sentences into algebraic equations to find unknowns.
Pattern Recognition
Multi-step equations train you to see structure and plan a solution path.

The Golden Rule of Equations

Whatever you do to one side of an equation, you must do to the other side. This preserves the balance and keeps the equation true.

\(x + 5 = 12\)
Subtract 5 from both sides
\(x = 7\)
\(3x = 18\)
Divide both sides by 3
\(x = 6\)
Inverse Operations

Inverse Operations

Operation Inverse Example
Addition (\( +\,a \)) Subtraction (\( -\,a \)) \( x + 5 = 9 \;\Rightarrow\; x = 4 \)
Subtraction (\( -\,a \)) Addition (\( +\,a \)) \( x - 3 = 7 \;\Rightarrow\; x = 10 \)
Multiplication (\( a \cdot \)) Division (\( \div\,a \)) \( 4x = 20 \;\Rightarrow\; x = 5 \)
Division (\( \div\,a \)) Multiplication (\( a \cdot \)) \( \frac{x}{2} = 6 \;\Rightarrow\; x = 12 \)

One-Step vs Multi-Step

One-Step
Requires one inverse operation to isolate the variable.

Example: \( x + 7 = 15 \)
Subtract 7: \( x = 8 \)
Multi-Step
Requires two or more inverse operations, often with distributing or combining like terms.

Example: \( 2x + 3 = 11 \)
Subtract 3, then divide by 2: \( x = 4 \)

How to Solve Any Equation

1
Simplify each side: Distribute parentheses and combine like terms.
2
Collect variable terms: Move all variable terms to one side using addition or subtraction.
3
Collect constants: Move all constant terms to the opposite side.
4
Isolate the variable: Divide (or multiply) both sides by the coefficient.
5
Check your answer: Substitute back into the original equation to verify.

Example 1: One-Step Equation

Solve: \( x - 9 = 14 \)

Given:
\(x - 9 = 14\)
Add 9:
\(x - 9 + 9 = 14 + 9\)
Solution:
\(x = 23\)

Example 2: Two-Step Equation

Solve: \( 3x + 7 = 22 \)

Given:
\(3x + 7 = 22\)
Subtract 7:
\(3x = 15\)
Divide by 3:
\(x = 5\)
Check:
\(3(5) + 7 = 15 + 7 = 22\) ✓

Example 3: Distribution Required

Solve: \( 2(x + 4) - 3 = 15 \)

1
Distribute: \( 2x + 8 - 3 = 15 \)
2
Combine like terms: \( 2x + 5 = 15 \)
3
Subtract 5: \( 2x = 10 \)
4
Divide by 2: \( x = 5 \)

Example 4: Variables on Both Sides

Solve: \( 5x - 2 = 3x + 10 \)

1
Subtract \( 3x \) from both sides: \( 2x - 2 = 10 \)
2
Add 2 to both sides: \( 2x = 12 \)
3
Divide by 2: \( x = 6 \)

Example 5: Equations with Fractions

Solve: \( \dfrac{x}{4} + 3 = 7 \)

1
Subtract 3: \( \dfrac{x}{4} = 4 \)
2
Multiply by 4: \( x = 16 \)

Common Mistakes

✕ Forgetting Both Sides
Adding or subtracting on only one side of the equation breaks the balance. Always apply operations to both sides.
✕ Distribution Errors
In \( 3(x - 4) \), you must multiply both terms: \( 3x - 12 \), not \( 3x - 4 \).
✕ Sign Errors
Moving a negative term to the other side means it becomes positive. Watch for \( -(-a) = +a \).

Summary Comparison

Feature One-Step Multi-Step
Operations needed \( 1 \) \( \geq 2 \)
Distribution Never Sometimes
Combining like terms Never Often
Variables on both sides No Possible
Example \( x + 5 = 12 \) \( 4(x - 1) = 2x + 6 \)
Difficulty Basic Intermediate

Question 1 of 5

Solve for \( x \): \( x + 13 = 27 \)

Question 2 of 5

Solve for \( x \): \( 4x - 9 = 23 \)

Question 3 of 5

Solve for \( x \): \( 3(x + 2) = 21 \)

Question 4 of 5

Solve for \( x \): \( 7x + 4 = 3x + 24 \)

Question 5 of 5

Solve for \( x \): \( \dfrac{2x - 6}{3} = 4 \)