Unit 2 · Lesson 1 of 6

Solving linear equations

About 20-25 minutes · 5 steps · one idea: do the same thing to both sides

You'll be able to: solve an equation like \(3(x-4) - x = 2 - (x+5)\) step by step, and check your answer.

  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. Answer from memory first. The first one is a skill you will use in every equation today.

Simplify \(2(3x - 4) - (x - 6)\).

Distribute: \(2(3x-4) = 6x - 8\), and \(-(x - 6) = -x + 6\).

Combine like terms: \(6x - x = 5x\) and \(-8 + 6 = -2\).

Answer: \(5x - 2\).

Multiply \(\dfrac{2}{3} \cdot \dfrac{9}{4}\). Simplify your answer.

Cancel first: 2 and 4 share a 2, and 3 and 9 share a 3. That leaves \(\dfrac{1}{1} \cdot \dfrac{3}{2}\).

Answer: \(\dfrac{3}{2}\).

Step 2 of 5The idea

An equation is like a balanced scale. Whatever you do to one side, you must do to the other side, and it stays balanced.

To solve it, you get the variable alone by undoing what was done to it. Adding is undone by subtracting, and multiplying is undone by dividing.

The steps for a linear equation:

  1. Simplify each side: distribute, then combine like terms.
  2. Move the variable terms to one side (add or subtract a term on both sides).
  3. Move the plain numbers to the other side.
  4. Divide both sides by the number in front of the variable.
  5. Check: put your answer into the original equation.

If the variable disappears: a true statement like \(6 = 6\) means every number works (an identity, answer "all"). A false one like \(6 = 1\) means no number works (a contradiction, answer "none").

Why this shows up on the placement test: nearly every later topic (lines, systems, quadratics, logs) ends with "now solve a linear equation", so this step has to be automatic.

Step 3 of 5Worked examples

Example 1: \(5x - 7 = 2x + 8\)

  1. Subtract \(2x\) from both sides: \(3x - 7 = 8\). Now the variable is only on the left.
  2. Add 7 to both sides: \(3x = 15\). Adding 7 undoes the \(-7\).
  3. Divide both sides by 3: \(x = 5\).
  4. Check: left side \(5(5) - 7 = 18\), right side \(2(5) + 8 = 18\). They match.

Example 2: \(3(x-4) - x = 2 - (x+5)\)

  1. Simplify the left side: \(3x - 12 - x = 2x - 12\).
  2. Simplify the right side: \(2 - x - 5 = -x - 3\). The minus in front of \((x+5)\) changes both signs.
  3. Now solve \(2x - 12 = -x - 3\). Add \(x\) to both sides: \(3x - 12 = -3\).
  4. Add 12 to both sides: \(3x = 9\). Divide by 3: \(x = 3\).
  5. Check: left \(3(3-4) - 3 = -3 - 3 = -6\), right \(2 - (3+5) = 2 - 8 = -6\). They match.

Step 4 of 5Practice

Work on paper, type your answer, and press Check. If you miss twice, open the worked solution. Then mark how it went.

Solve \(3x - 4 = 17\).

Add 4 to both sides: \(3x = 21\).

Divide by 3: \(x = 7\).

Check: \(3(7) - 4 = 17\). Answer: \(x = 7\).

Solve \(-5x + 2 = 27\).

Subtract 2 from both sides: \(-5x = 25\).

Divide by \(-5\): \(x = -5\).

Check: \(-5(-5) + 2 = 25 + 2 = 27\). Answer: \(x = -5\).

Solve \(7x + 3 = 4x - 9\).

Subtract \(4x\) from both sides: \(3x + 3 = -9\).

Subtract 3: \(3x = -12\). Divide by 3: \(x = -4\).

Check: \(7(-4) + 3 = -25\) and \(4(-4) - 9 = -25\). Answer: \(x = -4\).

Solve \(2(3x - 1) = 4x + 10\).

Distribute: \(6x - 2 = 4x + 10\).

Subtract \(4x\): \(2x - 2 = 10\). Add 2: \(2x = 12\). Divide by 2: \(x = 6\).

Check: \(2(18 - 1) = 34\) and \(24 + 10 = 34\). Answer: \(x = 6\).

Solve \(5 - 2(x + 4) = 3x + 7\).

Distribute the \(-2\): \(5 - 2x - 8 = 3x + 7\), so \(-2x - 3 = 3x + 7\).

Subtract \(3x\): \(-5x - 3 = 7\). Add 3: \(-5x = 10\). Divide by \(-5\): \(x = -2\).

Check: \(5 - 2(2) = 1\) and \(3(-2) + 7 = 1\). Answer: \(x = -2\).

Solve \(3(x + 2) - x = 2(x + 3)\).

Left side: \(3x + 6 - x = 2x + 6\). Right side: \(2x + 6\).

Subtract \(2x\) from both sides: \(6 = 6\). The variable disappeared and the statement is true.

Answer: every number works (all real numbers). This is an identity.

Solve \(4x - (x - 5) = 3(x + 1)\).

Left side: \(4x - x + 5 = 3x + 5\). Right side: \(3x + 3\).

Subtract \(3x\) from both sides: \(5 = 3\). The variable disappeared and the statement is false.

Answer: no solution. This is a contradiction.

Which value is the solution of \(6 - 4(x - 2) = 2x - 4\)?

  1. \(x = \tfrac{1}{3}\)
  2. \(x = 3\)
  3. \(x = -9\)
  4. \(x = -3\)

Distribute the \(-4\): \(6 - 4x + 8 = 2x - 4\), so \(14 - 4x = 2x - 4\).

Add \(4x\) to both sides: \(14 = 6x - 4\). Add 4: \(18 = 6x\). Divide by 6: \(x = 3\). Answer (b).

Check: \(6 - 4(1) = 2\) and \(2(3) - 4 = 2\). Choice (a) comes from writing \(-4(x-2)\) as \(-4x - 8\).

Step 5 of 5Wrap-up

You can now solve a linear equation by simplifying each side, gathering the variable on one side, dividing, and checking, and you can spot when the answer is "all" or "none".

Read or watch next: OpenStax Intermediate Algebra 2e, 2.1 Use a General Strategy to Solve Linear Equations (free textbook, with more practice at the end of the section).

Also good: Khan Academy: Algebra 1: Linear equations with variables on both sides (short videos). Quick rules: Equations and inequalities cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "How do I decide which side to move the x terms to?" It knows this course and can explain things another way.