Training Linear Equations and Inequalities Solving One- and Two-Step Equations
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Solving One- and Two-Step Equations

22 min Linear Equations and Inequalities

Solving Linear Equations

An equation is a statement that two expressions are equal. Solving an equation means finding the value of the unknown variable that makes the statement true. One-step and two-step equations are the simplest types, requiring just one or two inverse operations to isolate the variable.

The golden rule of equation solving is balance: whatever you do to one side, you must do to the other. This principle — rooted in the properties of equality — will carry you through every equation you ever solve, from basic algebra through differential equations.

This lesson teaches you to solve one-step equations (using addition, subtraction, multiplication, or division) and two-step equations (which combine two operations), building the mechanical fluency that more complex problems demand.

Definition

A linear equation in one variable has the form $ax + b = c$. The solution is the value of $x$ that makes the equation true.

Strategy

Isolate the variable using inverse operations. Whatever you do to one side, do to the other.

One-Step Equations

Example 1

$x + 7 = 12$

  1. Subtract 7:
  2. $x = 12 - 7 = 5$.
  3. Check: $5 + 7 = 12$ ✓
Example 2

$4x = 20$

  1. Divide by 4:
  2. $x = 5$.
  3. Check: $4(5) = 20$ ✓
Example 3

$\dfrac{x}{3} = 9$

  1. Multiply by 3:
  2. $x = 27$.
  3. Check: $\dfrac{27}{3} = 9$ ✓
Example 4

$x - 15 = -8$

  1. Add 15:
  2. $x = -8 + 15 = 7$.

Two-Step Equations

Undo addition/subtraction first, then multiplication/division.

Example 5

$3x + 7 = 22$

  1. Subtract 7:
  2. $3x = 15$
  3. Divide by 3: $x = 5$

Check: $3(5) + 7 = 22$ ✓

Example 6

$\dfrac{x}{4} - 3 = 5$

  1. Add 3:
  2. $\dfrac{x}{4} = 8$
  3. Multiply by 4: $x = 32$
Example 7

$-5x + 2 = -18$

  1. Subtract 2:
  2. $-5x = -20$
  3. Divide by $-5$: $x = 4$
Interactive Explorer: One- & Two-Step Equations
Equation: 3x + 7 = 19
Step 1 (subtract b): 3x = 12
Step 2 (divide by a): x = 4
Check: 3(4) + 7 = 19 ✓

Practice Problems

1. $x + 9 = 14$
2. $7x = -42$
3. $x - 11 = 3$
4. $\dfrac{x}{5} = -4$
5. $2x + 5 = 17$
6. $\dfrac{x}{3} - 7 = 1$
7. $-4x + 9 = 1$
8. $6x - 10 = 20$
9. $\dfrac{x}{-2} + 4 = 10$
10. $-3x - 7 = 8$
11. $15 - x = 9$
12. $0.5x + 3 = 7$
13. $\dfrac{2x}{3} = 10$
14. $-x = 12$
15. $8 - 3x = -1$
Show Answer Key

1. $x = 5$

2. $x = -6$

3. $x = 14$

4. $x = -20$

5. $x = 6$

6. $x = 24$

7. $x = 2$

8. $x = 5$

9. $x = -12$

10. $x = -5$

11. $x = 6$

12. $x = 8$

13. $x = 15$

14. $x = -12$

15. $x = 3$

Multi-Step Equations