Linear Equations

Grade 8 · algebra · 67 practice problems · read aloud

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Linear Equations: Finding the Unknown

A linear equation is like a balance scale. It shows that two expressions are equal, often with one unknown value (like x). We use them to solve real-world problems, like calculating budgets, determining speeds, or figuring out how many items you can buy. 🧮

How to Solve a One-Step Equation

The goal is to get the variable (e.g., x) by itself on one side of the equals sign. You do this by performing the inverse (opposite) operation to both sides.

Example 1: Solve for x: x + 7 = 15

  1. Identify the operation: The 7 is being added to x.
  2. Do the inverse: Subtract 7 from both sides.
    x + 7 - 7 = 15 - 7
  3. Simplify: x = 8

How to Solve a Two-Step Equation

When you have two operations, you must "undo" them in the reverse order of PEMDAS (undo addition/subtraction first, then multiplication/division).

Example 2: Solve for x: 3x - 5 = 16

  1. Undo subtraction: Add 5 to both sides.
    3x - 5 + 5 = 16 + 5 → 3x = 21
  2. Undo multiplication: Divide both sides by 3.
    3x / 3 = 21 / 3
  3. Simplify: x = 7

Common Mistakes to Avoid ⚠️

  • Forgetting to balance both sides: Whatever you do to one side of the equation, you MUST do to the other.
  • Incorrect inverse operations: Addition undoes subtraction, and multiplication undoes division.
  • Misapplying the distributive property: Remember that a number outside parentheses multiplies everything inside (e.g., 2(x + 3) = 2x + 6).

Tips & Tricks

  • Think of the equation as a balance scale that must always stay level.
  • Check your answer! Substitute your solution back into the original equation to see if it makes a true statement.
  • For two-step equations, remember the mantra: "Add or subtract first, then multiply or divide."

How to Practice

Start simple and build your skills! Try these:

  • Solve: m / 4 = 3
  • Solve: 2y + 8 = 20
  • Solve: 5(n - 2) = 25

Create your own problems or use online practice websites. The key is consistent practice! 💪

Practice problems

6 of the 67, worked through step by step — try them before opening the answer.

1 2(3x - 5) + 7 = 4x + 9 = ?

Hint: First distribute any multiplication across terms in parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: 6

  1. Distribute the 2: 2(3x - 5) + 7 = 6x - 10 + 7
  2. Combine like terms on the left: 6x - 3 = 4x + 9
  3. Subtract 4x from both sides: 6x - 4x - 3 = 9 → 2x - 3 = 9
  4. Add 3 to both sides: 2x = 12
  5. Divide both sides by 2: x = 6

The answer is 6.

2 3(2x - 5) + 7 = 4(x + 3) - 2

Hint: Distribute any coefficients outside parentheses first, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 7

To solve linear equations with variables on both sides, first use the distributive property to eliminate parentheses. Then combine like terms on each side. Finally, use inverse operations to isolate the variable term on one side and the constant terms on the other side.

3 3(2x - 5) + 7 = 4(x + 3) - 2 = ?

Hint: Begin by distributing any coefficients outside parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 6

  1. Expand both sides** Left side: 3(2x - 5) + 7 = 3 * 2x - 3 * 5 + 7 = 6x - 15 + 7 = 6x - 8 Right side: 4(x + 3) - 2 = 4 * x + 4 * 3 - 2 = 4x + 12 - 2 = 4x + 10 So the equation becomes: 6x - 8 = 4x + 10 --- **
  2. Move x terms to one side** Subtract 4x from both sides: 6x - 8 - 4x = 4x + 10 - 4x 2x - 8 = 10 --- **
  3. Move constant terms to the other side** Add 8 to both sides: 2x - 8 + 8 = 10 + 8 2x = 18 --- **
  4. Solve for x** Divide both sides by 2: x = 18 / 2 x = 6 --- **
  5. Check** Left side: 3(2*6 - 5) + 7 = 3(12 - 5) + 7 = 3*7 + 7 = 21 + 7 = 28 Right side: 4(6 + 3) - 2 = 4*9 - 2 = 36 - 2 = 28 Both sides equal, so the solution is correct. --- **Final answer:** x = 6

Let's solve the equation step by step. We have: 3(2x - 5) + 7 = 4(x + 3) - 2 --- **

4 2(3x - 5) + 4 = 3(x + 2) - 1 = ?

Hint: Start by distributing any numbers outside parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 3

  1. Distribute on both sides 2(3x - 5) + 4 = 3(x + 2) - 1 6x - 10 + 4 = 3x + 6 - 1
  2. Combine like terms on each side 6x - 6 = 3x + 5
  3. Get all x terms on one side by subtracting 3x from both sides 6x - 6 - 3x = 3x + 5 - 3x 3x - 6 = 5
  4. Isolate the x term by adding 6 to both sides 3x - 6 + 6 = 5 + 6 3x = 11
  5. Solve for x by dividing both sides by 3 3x/3 = 11/3 x = 11/3
  6. Check the solution 2(3*(11/3) - 5) + 4 = 3((11/3) + 2) - 1 2(11 - 5) + 4 = 3(11/3 + 6/3) - 1 2(6) + 4 = 3(17/3) - 1 12 + 4 = 17 - 1 16 = 16 ✓

The answer is x = 11/3.

5 2(3x - 5) + 7 = 4(x + 1) - 3 = ?

Hint: First distribute any numbers outside parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: 2

  1. Distribute on both sides: 2(3x - 5) + 7 = 6x - 10 + 7 and 4(x + 1) - 3 = 4x + 4 - 3
  2. Combine like terms on both sides: 6x - 3 = 4x + 1
  3. Subtract 4x from both sides: 6x - 4x - 3 = 1 → 2x - 3 = 1
  4. Add 3 to both sides: 2x = 4
  5. Divide both sides by 2: x = 2

The answer is 2.

6 4(2x - 3) + 7 = 3(3x + 1) - 5 = ?

Hint: Begin by distributing the numbers outside the parentheses to eliminate them, then combine like terms on each side before isolating the variable term.

Show the answer

Answer: -3

  1. Distribute on both sides Left side: 4(2x - 3) = 8x - 12 Right side: 3(3x + 1) = 9x + 3 Equation becomes: 8x - 12 + 7 = 9x + 3 - 5
  2. Combine like terms on each side Left side: 8x - 5 Right side: 9x - 2 Equation becomes: 8x - 5 = 9x - 2
  3. Move variable terms to one side Subtract 9x from both sides: 8x - 5 - 9x = 9x - 2 - 9x Simplifies to: -x - 5 = -2
  4. Move constant terms to the other side Add 5 to both sides: -x - 5 + 5 = -2 + 5 Simplifies to: -x = 3
  5. Solve for x Multiply both sides by -1: -x × (-1) = 3 × (-1) x = -3

The answer is -3.

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