Transposing terms

1Summary 重點摘要

Transposing = crossing the river with a sign change: + becomes −, − becomes +, × becomes ÷, ÷ becomes ×. Gather x on one side, numbers on the other.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What transposition isTransposition moves a term from one side of the equals sign to the other so that like terms gather together. In 2x + 3 = 7, moving + 3 across gives 2x = 7 − 3, with all variable terms on the left and constants on the right — one step closer to the answer. Transposition is really shorthand for 'add or subtract the same quantity on both sides': subtracting 3 everywhere has exactly the same effect as moving + 3 across and flipping its sign. It makes the written solution far shorter than the balance method.
2. 2. The rule: change the signThe iron rule: crossing the equals sign forces a sign change. A + 3 on the left becomes − 3 on the right, and a − 5 becomes + 5; signs always travel with their terms. The rule cannot be bent, because transposition is subtraction (or addition) of the term on both sides, which necessarily flips its sign. Before writing, circle the terms to be moved, recite 'cross the sign, change the sign', and copy each one to the other side with its new sign — omissions then become rare.
3. 3. Move constants to the rightThe first exercise type moves constants to the right. For 3x + 2 = 11: move + 2 across as − 2, giving 3x = 11 − 2 = 9, then divide both sides by 3 to get x = 3. Similarly 2x + 3 = 7 gives 2x = 7 − 3 = 4, hence x = 2. The routine is fixed: move the constant first, divide by the coefficient afterwards. For equations carrying a minus, such as 3x − 2 = 11, the − 2 crosses as + 2, giving 3x = 13; a non-integer result is perfectly normal.
4. 4. Move variables to the leftWhen x appears on both sides, as in 2x + 3 = x + 7, move the right-hand + x to the left as − x: 2x − x + 3 = 7, which combines to x + 3 = 7, and the usual moves finish it at x = 4. The convention is variables on the left, constants on the right; answers then look uniform and stay easy to check. If the right-hand x term carries a minus, as in x − 4, it crosses to the left as + 4 — exactly the same rule as for constants, applied without exception.
5. 5. Variables and constants on both sidesThe full type has variables and constants on both sides, such as 5x − 2 = 3x + 6: first move 3x left and − 2 right, giving 5x − 3x = 6 + 2; second, combine like terms to get 2x = 8; third, divide by 2 to get x = 4. The worked example 5x + 3 = 2x + 15 follows the same route: 5x − 2x = 15 − 3, so 3x = 12 and x = 4. The motto: variables left, constants right, signs flipped on the way, coefficients handled last — a fixed order keeps even long questions orderly.
6. 6. CheckingAfter solving, check by substituting into both sides. For 5x − 2 = 3x + 6 with x = 4: left = 5(4) − 2 = 18, right = 3(4) + 6 = 18, and equality confirms the answer; the worked example passes with 23 on each side. Checking is especially good at catching missed sign changes, because a wrong sign almost always breaks the equality. Write the two lines 'left =' and 'right =' explicitly and accept the answer only when the values agree.
7. 7. Common mistakesThe biggest trap is transposing without changing the sign: in 2x + 3 = 7, the wrong move is 2x = 7 + 3, while the correct move is 2x = 7 − 3. Other frequent slips include moving variables and constants in the wrong directions so the equation grows messier, dividing by the coefficient before combining like terms, and skipping the check altogether. Protection comes from writing each transposition on its own line, inspecting every sign, and finishing with verification — with that routine, errors become nearly impossible.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Transposition: move term to other side
• Always change sign (+ ↔ −)
• Variables left, constants right
• After moving, combine like terms
• Check: substitute back