In this lesson, you will learn the basic methods for solving linear equations. You will use the “properties of equality” to find the value of the unknown x.
Properties of Equality
- An equation remains true when the same number is added to both sides
- An equation remains true when the same number is subtracted from both sides
- An equation remains true when both sides are divided by the same number (division by 0 is not allowed)
- An equation remains true when both sides are multiplied by the same number
Example 1 (Basic)
Solve 3x＋5＝14.
Solution:
3x＝14－5
3x＝9
x＝3
Solve 3x＋5＝14.
Solution:
3x＝14－5
3x＝9
x＝3
Rules for Transposing Terms
Moving a term from one side of an equation to the other while changing its sign is called transposing a term.
Example 2 (When x appears on both sides)
Solve 5x－3＝2x＋9.
Solution:
5x－2x＝9＋3
3x＝12
x＝4
Solve 5x－3＝2x＋9.
Solution:
5x－2x＝9＋3
3x＝12
x＝4
Example 3 (When parentheses are present)
Solve 3(x＋2)＝2(x－1)＋11.
Solution:
3x＋6＝2x－2＋11
3x－2x＝－2＋11－6
x＝3
Solve 3(x＋2)＝2(x－1)＋11.
Solution:
3x＋6＝2x－2＋11
3x－2x＝－2＋11－6
x＝3
Key Points
- When transposing a term, its sign changes（＋↔−）
- Collect the terms containing x on the left side and the numbers on the right side
- Expand the parentheses first
Practice Problems
- Solve 4x－7＝13.
- Solve 2(3x－1)＝4x＋8.
Answers and Explanations
- Answer:x＝5
Explanation: 4x＝13＋7＝20. x＝20÷4＝5. Check: 4×5－7＝13 ✓ - Answer:x＝5
Explanation: Expand the parentheses: 6x－2＝4x＋8. Transpose the terms: 6x－4x＝8＋2. 2x＝10. x＝5. Check: 2(3×5-1)＝28＝4×5＋8 ✓