Chapter 2.2 — Exercise 2.1 — Simple Equations

The Transposition Method — How It Works

Transposing a term means moving it across the equals sign, and the operation it performs flips as it moves: a number being added on one side becomes subtracted on the other, and a number multiplying one side becomes dividing on the other. The general pattern is worth holding in your head as you work through the sixteen questions: transpose the constant term first, then deal with the coefficient of the variable.

Once a candidate solution is found, substitute it straight back into the original equation and confirm LHS equals RHS — that check is not optional, since it's the only way to catch a sign that slipped during transposition.

Solutions at a Glance — All 16 Equations

• (i) 6m = 12 → m = 12/6 = 2
• (ii) 14p = −42 → p = −42/14 = −3
• (iii) −5y = 30 → y = 30/(−5) = −6
• (iv) −2x = −12 → x = −12/(−2) = 6
• (v) 34x = −51 → x = −51/34 = −3/2
• (vi) n/7 = −3 → n = −3 × 7 = −21
• (vii) 2x/3 = 18 → 2x = 54 → x = 27
• (viii) 3x + 1 = 16 → 3x = 15 → x = 5
• (ix) 3p − 7 = 0 → 3p = 7 → p = 7/3
• (x) 13 − 6n = 7 → −6n = −6 → n = 1
• (xi) 200y − 51 = 49 → 200y = 100 → y = 1/2
• (xii) 11n + 1 = 1 → 11n = 0 → n = 0
• (xiii) 7x − 9 = 16 → 7x = 25 → x = 25/7
• (xiv) 8x + 5/2 = 13 → 8x = 21/2 → x = 21/16
• (xv) 4x − 5/3 = 9 → 4x = 32/3 → x = 8/3
• (xvi) x + 4/3 = 3½ → x = 7/2 − 4/3 = 13/6

Worked Example — When the Coefficient Is a Fraction Too

Question (xiv), 8x + 5/2 = 13, mixes an integer coefficient with a fractional constant, which is exactly where transposition order matters most. Transpose 5/2 to the right first: 8x = 13 − 5/2 = 26/2 − 5/2 = 21/2. Only now transpose the 8: x = 21/2 ÷ 8 = 21/16. Checking: 8 × (21/16) + 5/2 = 21/2 + 5/2 = 26/2 = 13, which matches the original right-hand side exactly.

Why Transposition Actually Works

Transposition can look like a trick — move a term, flip its sign, done — but it's really just a shortcut for something more fundamental: whatever you do to one side of an equation, you must do to the other side too, to keep the two sides equal. Take 6m = 12. Dividing both sides by 6 gives 6m/6 = 12/6, which simplifies to m = 2. Writing "transpose 6 to the RHS as division" is just a faster way of describing that same both-sides operation, without re-deriving it from scratch every time. Seeing transposition this way also explains why the sign or operation flips: subtracting the same term from both sides of a + b = c turns addition into subtraction on the side it's removed from, and dividing both sides by the same coefficient turns multiplication into division.

Grouping the Sixteen Questions by Type

Looking across all sixteen equations, they fall into a few recognisable shapes: a variable directly multiplied by a whole number (i–v), a variable divided by a number (vi–vii), a variable with one extra constant term added or subtracted (viii–xiii), and a variable combined with a fractional constant (xiv–xvi). Recognising which shape an equation matches before starting tells you immediately how many transposition steps it will take — a single multiplication needs one step, while a coefficient plus a fractional constant needs two. This kind of pattern-spotting is worth doing deliberately at first, since it's exactly the instinct that later lets you glance at a more complicated equation and know, before writing a single line, roughly how many moves solving it will take.

Where Sign Errors Creep In

• Negative coefficients. In −5y = 30, dividing by −5 (not 5) gives y = −6 — the negative sign on the coefficient carries through to the division.
• Two negatives dividing. In −2x = −12, a negative divided by a negative gives a positive result: x = 6, not −6.
• Mixed numbers on the RHS. A value like 3½ needs converting to an improper fraction (7/2) before any transposing or combining happens — operating on a mixed number directly is a common source of arithmetic slips.
• Fractional coefficients. When the variable's coefficient is itself a fraction, like 2x/3, clear the denominator first (multiply both sides by 3) before isolating the coefficient of x.

One More Worked Example, Chosen for Its Sign Changes

Question (xi), 200y − 51 = 49, is worth walking through slowly because both transposition steps involve a change of sign. First, transpose −51 to the right, which becomes +51: 200y = 49 + 51 = 100. Then transpose 200, which divides rather than multiplies on the other side: y = 100/200 = 1/2. Checking: 200 × (1/2) − 51 = 100 − 51 = 49, which matches the original right-hand side. Two transpositions, two sign or operation changes, and a fractional answer at the end — this single equation touches almost every idea in this exercise at once. Working through it slowly once, rather than rushing to the answer, is exactly the practice that makes the faster equations in this list feel automatic, since every one of them uses the same two moves — transpose, then simplify — in a shorter combination — often just one of the two, when the other side is already a bare number, and both when a coefficient and a constant both need moving — the shape of the equation tells you which before you write a single step — a habit worth carrying into every equation in the exercises that follow.

The Base Technique for Everything Else in This Chapter

The transposition skill built here is the base technique for the rest of Chapter 2, where equations grow more demanding — variables appearing on both sides, brackets that need expanding first, and fractional equations that need clearing before transposing. It's the same technique used whenever a word problem is translated into an equation and then solved. For the underlying concepts this exercise assumes, revisit the Introduction to Linear Equations. These skills connect directly to algebraic expressions and continue on to Exercise 2.2, which puts this same technique to work on real-world word problems.