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Algebra · Formulae & rearranging

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Changing the subject when it appears twice

Why a subject that appears twice must be collected and FACTORISED out, how clearing a fraction first exposes both copies, and the four-step routine that untangles any of these.

Algebra · Formulae & rearranging

Changing the subject when it appears twice

Why a subject that appears twice must be collected and FACTORISED out, how clearing a fraction first exposes both copies, and the four-step routine that untangles any of these.

Why it works

When the wanted letter appears twice, no amount of one-step peeling can finish the job — moving things around always leaves an xx on each side. The escape is factorising: two xx-terms collect into ONE xx times a bracket, and then a single division sets it free.

The four-step routine:
  1. Clear any fractions (multiply by the denominator) — this exposes
every copy of the subject.
  1. Expand whatever brackets appear.
  2. Collect the subject-terms on one side, everything else on the other.
  3. Factorise the subject out, and divide by the bracket.
Watch it run on y=2x+3x−1y = \dfrac{2x + 3}{x - 1}, making xx the subject:

y(x−1)=2x+3    ⇒    xy−y=2x+3    ⇒    xy−2x=y+3y(x - 1) = 2x + 3 \;\;\Rightarrow\;\; xy - y = 2x + 3 \;\;\Rightarrow\;\; xy - 2x = y + 3 ⇒    x(y−2)=y+3    ⇒    x=y+3y−2.\Rightarrow\;\; x(y - 2) = y + 3 \;\;\Rightarrow\;\; x = \frac{y + 3}{y - 2}.

Why factorising is forced. At xy−2x=y+3xy - 2x = y + 3 the left side is a sum of two xx-terms; dividing by yy or by 2 alone would still leave xx in two places. x(y−2)x(y - 2) is the only shape with xx appearing once — that's what "take out the common factor" buys.

The answer must contain NO copy of the subject. If your final line has xx on both sides — x=2x+3+yyx = \frac{2x + 3 + y}{y} style — the rearrangement isn't finished, however true the equation is. That's the tell to go back and factorise.

Sign discipline at step 3. Choosing which side the xx-terms go decides the signs: from pm+4p=3−2mpm + 4p = 3 - 2m, bringing −2m-2m across gives pm+2m=3−4ppm + 2m = 3 - 4p — each crossing term flips. A wrong sign here poisons the bracket, so check the factorised line by expanding it back.

Same routine, simpler shapes. ax+7=bx+cax + 7 = bx + c needs no fraction-clear: collect (ax−bx=c−7ax - bx = c - 7), factorise (x(a−b)=c−7x(a - b) = c - 7), divide. Done.