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Linear Expressions
Add, Subtract, Factor

Master the art of combining like terms, distributing coefficients, and factoring out the GCF from linear expressions with rational coefficients.

7th Grade Math

What Is a Linear Expression?

A linear expression is a mathematical phrase built from variables (like \(x\)), constants (like 7), and operations (+, −, ×). The variable never has an exponent greater than 1.

Term
A single piece: a number, a variable, or a number times a variable. Example: \(3x\), \(-5\), \(\frac{1}{2}y\)
Coefficient
The number multiplied by a variable. In \(4x\), the coefficient is 4.
Constant
A term with no variable attached. In \(3x + 7\), the constant is 7.

Anatomy of a Linear Expression

\(3x + 5y - 7\)
term coefficient: 3 term coefficient: 5 constant value: −7

What Are Like Terms?

Like terms share the exact same variable part. Only the coefficients can differ. You can only add or subtract like terms together.

\(3x\)  and  \(7x\)
✓ Like terms Same variable: x
\(3x\)  and  \(3y\)
✕ Not like terms Different variables

The Distributive Property

The distributive property connects multiplication and addition. It lets you expand (multiply into parentheses) and factor (pull a common factor out).

\(a(b + c) = ab + ac\)
Expand → ← Factor
\(2(x + 4) = 2x + 8\)

Core Rule: Combining Like Terms

\(5x\)
 +
\(3x\)
 =
\(8x\)
 coeff: 5 coeff: 3
\(5 + 3 = 8\)
 The variable x stays the same: only coefficients change.

Core Rule: Factoring with the GCF

Before
\(6x + 12\)
 → After (GCF = 6)
\(6(x + 2)\)
Check: \(6 \times x + 6 \times 2 = 6x + 12\) ✓

Expanding vs. Factoring

These are inverse operations: expanding multiplies a factor across terms, while factoring pulls a common factor out. They undo each other.

Expanding Factoring
Direction Parentheses → no parentheses No parentheses → parentheses
Operation Distribute (multiply) Find GCF and divide
Example \(3(x+4) = 3x+12\) \(3x+12 = 3(x+4)\)
When used To simplify before combining To simplify a final answer

How to Simplify Step by Step

1
Distribute any factors into parentheses using the distributive property.
2
Identify like terms: group variable terms together and constants together.
3
Combine like terms: add or subtract the coefficients.
4
Factor (if asked): find the GCF and rewrite in factored form.
5
Check your work: expand your answer to verify it matches the original.

Example 1: Adding Expressions

Simplify: \((4x + 3) + (2x - 5)\)

1
Remove parentheses: \(4x + 3 + 2x - 5\)
2
Group like terms: \((4x + 2x) + (3 - 5)\)
3
Combine: \(6x + (-2) = 6x - 2\)

Example 2: Subtracting Expressions

Simplify: \((7x + 9) - (3x + 4)\)

1
Distribute the negative: \(7x + 9 - 3x - 4\)
2
Group like terms: \((7x - 3x) + (9 - 4)\)
3
Combine: \(4x + 5\)

Example 3: Expand and Combine

Simplify: \(3(2x - 1) + 4(x + 5)\)

1
Distribute 3: \(3 \cdot 2x + 3 \cdot (-1) = 6x - 3\)
2
Distribute 4: \(4 \cdot x + 4 \cdot 5 = 4x + 20\)
3
Combine all terms: \(6x - 3 + 4x + 20 = 10x + 17\)

Example 4: Factoring a Linear Expression

Factor completely: \(8x + 20\)

1
Find the GCF of 8 and 20. Factors of 8: 1, 2, 4, 8. Factors of 20: 1, 2, 4, 5, 10, 20. GCF = 4.
2
Divide each term by 4: \(8x \div 4 = 2x\) and \(20 \div 4 = 5\).
3
Write in factored form: \(4(2x + 5)\).

Example 5: Rational Coefficients

Simplify: \(\frac{1}{2}(6x - 4) + \frac{3}{4}x\)

1
Distribute \(\frac{1}{2}\): \(\frac{1}{2} \cdot 6x + \frac{1}{2} \cdot (-4) = 3x - 2\)
2
Rewrite all terms: \(3x - 2 + \frac{3}{4}x\)
3
Combine \(x\)-terms: \(3x + \frac{3}{4}x = \frac{12}{4}x + \frac{3}{4}x = \frac{15}{4}x\)
4
Result: \(\frac{15}{4}x - 2\)

Common Mistakes to Avoid

✕ Forgetting to Distribute the Negative
In \((5x + 3) - (2x + 1)\), writing \(5x + 3 - 2x + 1\) instead of \(5x + 3 - 2x - 1\). The minus sign applies to every term in the second group.
✕ Combining Unlike Terms
Writing \(3x + 5 = 8x\). You cannot combine \(3x\) and \(5\) because they are not like terms. The correct result stays \(3x + 5\).
✕ Incomplete Factoring
Factoring \(12x + 18\) as \(2(6x + 9)\) instead of \(6(2x + 3)\). Always use the greatest common factor, not just any common factor.

Summary: Operations on Linear Expressions

Operation What You Do Example
Add Remove parentheses, combine like terms \((2x+3)+(x-1)=3x+2\)
Subtract Distribute negative, combine like terms \((5x+4)-(2x+1)=3x+3\)
Expand Multiply factor across each term \(3(x+2)=3x+6\)
Factor Find GCF, divide each term \(6x+9=3(2x+3)\)

Question 1 of 5

Simplify: \((3x + 7) + (5x - 2)\)

Question 2 of 5

Simplify: \((9x + 6) - (4x + 3)\)

Question 3 of 5

Expand and simplify: \(2(3x + 4) - 5x\)

Question 4 of 5

Factor completely: \(14x + 21\)

Question 5 of 5

Simplify: \(\frac{1}{3}(9x - 6) + 2x + 4\)