Simplifying Linear Expressions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Combine like terms in linear expressions.
  • Use the distributive property to expand expressions.
  • Rewrite expressions in equivalent simplified forms.
  • Avoid common mistakes when combining variables and constants.

Key Ideas

To simplify a linear expression means to rewrite it in its cleanest, most organized form by:

  • combining like terms
  • applying the distributive property
  • keeping negative signs under control
  • organizing variable terms first, constants second

Like Terms

Like terms share the same variable with the same exponent.

  • \(3x\) and \(-5x\) → like terms
  • \(2y\) and \(2x\) → not like terms
  • \(x\) and \(x^2\) → not like terms

Distributive Property

\[ a(b + c) = ab + ac \]

Standard Simplified Form

Write expressions as:

  • variable term(s) first
  • constant at the end

Example:
\[ 5x - 3x + 7 = 2x + 7 \]

Common Problem Types

1. Combining Like Terms

Example:
\[ 7x - 3x + 4 = 4x + 4 \]

2. Distributing Before Combining

Example:
\[ 2(x + 5) + 3 = 2x + 10 + 3 = 2x + 13 \]

3. Negative Signs Outside Parentheses

Example:
\[ -(3x - 5) = -3x + 5 \]

4. Multiple Variable Types

Example:
\[ 3x + 2y - x + y = 2x + 3y \]

5. Removing Parentheses Carefully

Example:
\[ (x + 4) - (2x - 1) = x + 4 - 2x + 1 = -x + 5 \]

Strategies

  • Circle or underline like terms before combining them.
  • Distribute first, then simplify.
  • Rewrite subtraction as adding a negative to avoid sign mistakes.
  • Lay out expressions vertically if the structure gets confusing.
  • Constantly track negative signs—this is where most errors occur.

Worked Examples

Example 1

Simplify:
\[ 6x - 4 + 3x + 7 \]

Step-by-step Solution:

  1. Combine variable terms → \(6x + 3x = 9x\)
  2. Combine constants → \(-4 + 7 = 3\)
  3. Put together → \(9x + 3\)

Answer: \(9x + 3\)

Example 2

Simplify:
\[ 5(2x - 1) - 3x \]

Step-by-step Solution:

  1. Distribute the 5 → \(10x - 5\)
  2. Add the outside term → \(10x - 5 - 3x\)
  3. Combine like terms → \(7x - 5\)

Answer: \(7x - 5\)

Example 3

Simplify:
\[ -(4x + 6) + 2(x - 3) \]

Step-by-step Solution:

  1. Distribute the negative → \(-4x - 6\)
  2. Distribute the 2 → \(2x - 6\)
  3. Combine like terms → \(-4x + 2x = -2x\)
  4. Combine constants → \(-6 - 6 = -12\)

Answer: \(-2x - 12\)

Example 4

Simplify:
\[ 3(x - 2) + 4(2 - x) \]

Step-by-step Solution:

  1. Expand both expressions → \(3x - 6 + 8 - 4x\)
  2. Combine variable terms → \(3x - 4x = -x\)
  3. Combine constants → \(-6 + 8 = 2\)

Answer: \(-x + 2\)

WarningCommon Mistakes
  • Combining unlike terms, such as \(x\) with \(x^2\) or \(x\) with constants.
  • Forgetting to distribute a negative sign across every term.
  • Only partially distributing (e.g., \(3(x + 2) = 3x + 2\)).
  • Changing signs incorrectly when removing parentheses.
  • Leaving expressions partially simplified.

Practice Problems

  1. \(8x - 3 + 2x + 5\)
  2. \(4(3x - 1) + 2x\)
  3. \(-(x - 7) + 5\)
  4. \(5y + 3 - 2(4y - 1)\)
  5. \((2x - 3) - (x + 4)\)

1. \(8x - 3 + 2x + 5\)
Step 1: Combine \(8x + 2x = 10x\)
Step 2: Combine \(-3 + 5 = 2\)
Answer: \(10x + 2\)

2. \(4(3x - 1) + 2x\)
Step 1: Distribute → \(12x - 4\)
Step 2: Add → \(12x - 4 + 2x\)
Step 3: Combine → \(14x - 4\)
Answer: \(14x - 4\)

3. \(-(x - 7) + 5\)
Step 1: Distribute negative → \(-x + 7\)
Step 2: Add +5 → \(-x + 12\)
Answer: \(-x + 12\)

4. \(5y + 3 - 2(4y - 1)\)
Step 1: Distribute → \(-8y + 2\)
Step 2: Combine variables → \(5y - 8y = -3y\)
Step 3: Combine constants → \(3 + 2 = 5\)
Answer: \(-3y + 5\)

5. \((2x - 3) - (x + 4)\)
Step 1: Distribute subtraction → \(2x - 3 - x - 4\)
Step 2: Combine → \(x - 7\)
Answer: \(x - 7\)

Summary

  • Combine like terms to simplify.
  • Distribute before combining when parentheses are present.
  • Track negative signs carefully—most errors happen there.
  • Write final expressions with variables first, constants last.
  • Combine like terms only when both the variable and exponent match.
  • Distribute before combining to avoid overlooking terms.
  • Rewrite subtraction as adding a negative to prevent sign mistakes.
  • Arrange expressions so variable terms come first and constants last.