Solving One-Step & Two-Step Equations

TipLearning Objectives

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

  • Understand what it means to solve a linear equation.
  • Solve one-step equations using inverse operations.
  • Solve two-step equations with one additional operation.
  • Check solutions by substitution to confirm correctness.

Key Ideas

A solution to an equation is a value that makes the equation true.

To solve an equation, we use inverse operations to isolate the variable on one side.

Common inverse pairs:

  • addition ↔︎ subtraction
  • multiplication ↔︎ division

A key principle:
Whatever you do to one side of the equation, do to the other to keep it balanced.

Common Problem Types

1. One-Step: Addition/Subtraction

Solve:
\[ x + 7 = 12 \]

Subtract 7 from both sides:
\[ x = 5 \]

2. One-Step: Multiplication/Division

Solve:
\[ 5x = 20 \]

Divide both sides by 5:
\[ x = 4 \]

3. Two-Step: \(ax + b = c\)

Solve:
\[ 3x + 4 = 10 \]

Step 1: subtract 4 → \(3x = 6\)
Step 2: divide by 3 → \(x = 2\)

4. Negative Coefficients

Solve:
\[ -4x = 20 \]

Divide both sides by \(-4\):
\[ x = -5 \]

Strategies

  • Use inverse operations to isolate the variable.
  • Write work in neat vertical steps to stay organized.
  • After solving, check your answer in the original equation.
  • Keep a close eye on negative signs—errors often happen there.

Worked Examples

Example 1

Solve:
\[ x - 9 = 4 \]

Solution:
1. Add 9: \(x = 13\)
2. Check: \(13 - 9 = 4\) ✓

Answer: \(13\)

Example 2

Solve:
\[ \frac{x}{6} = -3 \]

Solution:
1. Multiply both sides by 6: \(x = -18\)
2. Check: \(-18/6 = -3\) ✓

Answer: \(-18\)

Example 3

Solve:
\[ 2x - 5 = 9 \]

Solution:
1. Add 5 → \(2x = 14\)
2. Divide by 2 → \(x = 7\)
3. Check: \(2(7) - 5 = 9\) ✓

Answer: \(7\)

WarningCommon Mistakes
  • Doing a different operation to each side (breaking balance).
  • Forgetting to apply operations to every term on a side.
  • Losing track of negative signs when adding, subtracting, or dividing.
  • Skipping the solution check and missing simple errors.

Practice Problems

  1. Solve: \(x + 6 = 11\)
  2. Solve: \(9x = 45\)
  3. Solve: \(x - 4 = -1\)
  4. Solve: \(\dfrac{x}{5} = 7\)
  5. Solve: \(3x + 2 = 11\)
  6. Solve: \(-4x = 28\)

1. \(x + 6 = 11\)
Subtract 6 → \(x = 5\)

2. \(9x = 45\)
Divide by 9 → \(x = 5\)

3. \(x - 4 = -1\)
Add 4 → \(x = 3\)

4. \(\dfrac{x}{5} = 7\)
Multiply by 5 → \(x = 35\)

5. \(3x + 2 = 11\)
Subtract 2 → \(3x = 9\)
Divide by 3 → \(x = 3\)

6. \(-4x = 28\)
Divide by \(-4\) → \(x = -7\)

Summary

  • Solve equations using inverse operations to isolate the variable.
  • Keep both sides of an equation balanced.
  • Check solutions by substituting back into the original equation.
  • Be careful with negative signs—they matter at every step.
  • Undo addition/subtraction first, then undo multiplication/division.
  • Keep an eye on negative coefficients—they flip signs when dividing.
  • Use a balance-scale mindset: do the same thing to both sides.
  • Checking your answer catches almost every small slip.