Unit 1 · Lesson 6 of 6

Like terms and the distributive property

About 20-25 minutes · 5 steps · one idea: clear parentheses, then combine like terms

You'll be able to: simplify expressions like \(3(2x - 5) - 4(x - 1)\) by distributing and combining like terms.

  1. 1 Warm-up
  2. 2 The idea
  3. 3 Examples
  4. 4 Practice
  5. 5 Wrap-up

Step 1 of 5Warm-up

Two quick questions from earlier lessons. Try each one from memory before opening anything.

Evaluate \(x^2 - 3x\) when \(x = -2\).

\((-2)^2 - 3(-2) = 4 - (-6) = 4 + 6 = 10\).

\(\dfrac{2}{3} - \dfrac{3}{4}\)

The LCD is 12. \(\dfrac{2}{3} = \dfrac{8}{12}\) and \(\dfrac{3}{4} = \dfrac{9}{12}\). Tops: \(8 - 9 = -1\). Answer: \(-\dfrac{1}{12}\).

Step 2 of 5The idea

A term is one piece of an expression, separated by + or − signs. In \(4x^2 - 7x + 2\) the terms are \(4x^2\), \(-7x\), and \(2\).

The number in front of the letter is the coefficient: in \(-7x\) it is \(-7\). A term with no letter, like 2, is a constant.

Like terms have exactly the same letter part. \(3x\) and \(-5x\) are like terms. \(3x\) and \(3x^2\) are not.

Combine like terms. Add their coefficients and keep the letter part: \(3x - 5x = -2x\). A lone \(x\) has coefficient 1.

Distributive property. \(a(b + c) = ab + ac\): multiply the outside number by every term inside.

Distributing a negative. The minus sign goes to every term: \(-(x - 3) = -x + 3\) and \(-2(x - 4) = -2x + 8\).

To simplify: distribute first, then combine like terms.

Order of operations still applies. In \(7 - 2(x + 1)\), the 2 multiplies first. Do not do \(7 - 2\) first.

Why this shows up on the placement test: nearly every equation in later units starts by clearing parentheses and combining like terms, and "which expression is equivalent" questions test exactly this.

Step 3 of 5Worked examples

Example 1: \(5x^2 + 3x - 2x^2 - 7x + 4\)

  1. Group like terms, keeping each sign with its term: \(5x^2 - 2x^2\), then \(3x - 7x\), then \(4\).
  2. Combine: \(5x^2 - 2x^2 = 3x^2\) and \(3x - 7x = -4x\). (Add the coefficients: \(5 - 2 = 3\) and \(3 - 7 = -4\).)
  3. Result: \(3x^2 - 4x + 4\).

Example 2: \(3(2x - 5) - 4(x - 1)\)

  1. Distribute the 3: \(3 \cdot 2x = 6x\) and \(3 \cdot (-5) = -15\).
  2. Distribute the \(-4\), sign included: \(-4 \cdot x = -4x\) and \(-4 \cdot (-1) = +4\). (Negative times negative is positive.)
  3. Now you have \(6x - 15 - 4x + 4\).
  4. Combine: \(6x - 4x = 2x\) and \(-15 + 4 = -11\). Result: \(2x - 11\).

Step 4 of 5Practice

Simplify each expression completely. Type answers like 2x - 11 (use ^ for powers, like 3x^2), then press Check. If you miss twice, open the worked solution. Then mark how it went.

Simplify \(7x - 3 + 2x + 9\).

\(x\)-terms: \(7x + 2x = 9x\). Constants: \(-3 + 9 = 6\). Answer: \(9x + 6\).

Simplify \(4y^2 - 6y + y^2 + 2y\).

\(y^2\)-terms: \(4y^2 + y^2 = 5y^2\) (a lone \(y^2\) counts as \(1y^2\)). \(y\)-terms: \(-6y + 2y = -4y\). Answer: \(5y^2 - 4y\).

Simplify \(-(x - 8) - (2x + 5)\).

Flip every sign inside each set of parentheses: \(-x + 8 - 2x - 5\). Combine: \(-x - 2x = -3x\) and \(8 - 5 = 3\). Answer: \(-3x + 3\).

Simplify \(5 - 3(2x - 4)\).

Multiply first; the 3 goes with its minus sign: \(-3 \cdot 2x = -6x\) and \(-3 \cdot (-4) = +12\). So \(5 - 6x + 12\). Combine the constants: \(5 + 12 = 17\). Answer: \(17 - 6x\) (same as \(-6x + 17\)).

Simplify \(2(x + 6) + 5(x - 3)\).

Distribute: \(2x + 12 + 5x - 15\). Combine: \(2x + 5x = 7x\) and \(12 - 15 = -3\). Answer: \(7x - 3\).

Simplify \(4(3x - 2) - 2(5x - 7)\).

Distribute the 4: \(12x - 8\). Distribute the \(-2\): \(-10x + 14\). Combine \(12x - 8 - 10x + 14\): \(12x - 10x = 2x\) and \(-8 + 14 = 6\). Answer: \(2x + 6\).

Simplify \(3(x - 2y) - (4x - y)\).

Distribute: \(3x - 6y - 4x + y\). Combine the \(x\)-terms: \(3x - 4x = -x\). Combine the \(y\)-terms: \(-6y + y = -5y\). Answer: \(-x - 5y\).

Which expression is equivalent to \(6 - 2(x - 3)\)?

  1. \(12 - 2x\)
  2. \(4x - 12\)
  3. \(3 - 2x\)
  4. \(9 - 2x\)

Distribute the \(-2\): \(-2 \cdot x = -2x\) and \(-2 \cdot (-3) = +6\). So \(6 - 2x + 6 = 12 - 2x\). Answer: (a).

(b) does \(6 - 2\) first. (c) multiplies only the \(x\) and keeps \(-3\). (d) flips the sign of 3 but forgets to multiply it by 2.

Step 5 of 5Wrap-up

You can now simplify expressions by distributing (including a negative, sign and all) and then combining like terms. That finishes the lessons of Unit 1; the Unit 1 checkpoint is next.

Read or watch next: OpenStax Elementary Algebra 2e, 1.2 Use the Language of Algebra (like terms) and 1.9 Properties of Real Numbers (the distributive property).

Also good: Khan Academy: Combining like terms (short videos). Quick rules: Number skills cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "Why does \(-(x - 3)\) become \(-x + 3\) and not \(-x - 3\)?" It knows this course and can explain things another way.