Writing the properties of equality every time is difficult, so we will learn a faster way to solve equations. It is called transposition. With transposition, an equation can be solved simply by moving a term to the other side while changing its sign.
What Is Transposition?
Moving a term on one side of an equation to the opposite side while changing its sign is called transposition. This is a shortened form of the operation “adding or subtracting the same number from both sides.”
Example: x + 5 = 12 → x = 12 − 5 (When +5 is transposed to the right side, it becomes −5.)
- Move +5 on the left side to the right → its sign changes to −5
- Move −3 on the right side to the left → its sign changes to +3
Solve the following equations using transposition.
(1) x + 6 = 10 (2) x − 4 = 7 (3) 3x = 2x + 8
Answer:
(1) Transpose +6 to the right: x = 10 − 6 → x = 4
(2) Transpose −4 to the right: x = 7 + 4 → x = 11
(3) Transpose 2x on the right to the left: 3x − 2x = 8 → x = 8
How to Solve Linear Equations (Basic Pattern)
Solve a linear equation using the following steps.
① Collect the variable terms on the left side and the constant terms on the right side (transpose them)
② Simplify both sides (combine like terms)
③ Divide both sides by the coefficient of x (put the equation in the form x = )
Solve the following equation: 4x − 7 = x + 5
Answer:
① Transpose: move the x terms left and the constant terms right.
4x − x = 5 + 7
② Simplify: 3x = 12
③ Divide both sides by 3: x = 12 ÷ 3
x = 4
Solve the following equation: 5 − 2x = 3x − 10
Answer:
① Transpose: move the x terms left and the constant terms right.
−2x − 3x = −10 − 5
② Simplify: −5x = −15
③ Divide both sides by −5: x = (−15) ÷ (−5)
x = 3
- Transposition means moving a term to the opposite side while changing its sign. + ↔ −
- Linear equations: ① x terms left, constants right → ② simplify → ③ divide by the coefficient
- After solving, substitute the answer into the original equation to check it
- Even when the coefficient of x is negative, simply divide by the coefficient at the end
Practice Problems
- Solve the following: 5x = 3x + 14
- Solve the following: 6x − 9 = 4x + 3
- Solve the following: 8 − 3x = 2x − 7
Answers and Explanations
- Answer: x = 7
Explanation: Transpose 3x to the left: 5x − 3x = 14 → 2x = 14 → x = 7. - Answer: x = 6
Explanation: Transpose 4x to the left and −9 to the right: 6x − 4x = 3 + 9 → 2x = 12 → x = 6. - Answer: x = 3
Explanation: Transpose −3x to the left and 8 to the right: −3x − 2x = −7 − 8 → −5x = −15 → x = 3.