--- title: "Rearranging Formulae When the Subject Appears Twice" description: "Change the subject of a formula where the new subject shows up more than once: clear any fraction, collect those terms on one side, factorise the subject out, then divide." canonical: https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj source: https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Rearranging Formulae When the Subject Appears Twice Change the subject of a formula where the new subject shows up more than once: clear any fraction, collect those terms on one side, factorise the subject out, then divide. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj ## Ready when they can - Multiply through to clear a fraction before collecting terms - Gather every term containing the new subject on one side - Factorise the subject out of the collected terms and divide by the bracket ## Lesson: Rearranging Formulae When the Subject Appears Twice The last stop showed you how to handle algebraic fractions: clear the fraction, factorise, cancel whole factors, and watch the excluded values. This stop uses those same moves to change the subject of a formula when the new subject appears more than once. The method has four moves: clear any fraction, collect every term containing the new subject on one side, factorise the subject out, and divide by the bracket. *(drawing: The four moves, in the order they happen.)* Here is why the moves work. If x appears in two terms, you cannot just divide by x, because x is not a single factor yet. The trick is to make it one: collect every term containing x on one side and everything else on the other, then factorise x out. If x is inside a bracket, expand the bracket first. One more thing: the new denominator cannot be zero. In x = (2y + 1)/(y - 4), the value y = 4 would make the right side undefined, and the original formula has no solution there either. **Example.** Make x the subject of y = (2x + 5)/(x - 3). Clear the fraction by multiplying both sides by (x - 3): y(x - 3) = 2x + 5. Expand: xy - 3y = 2x + 5. Collect the x terms on the left: xy - 2x = 3y + 5. Factorise x out: x(y - 2) = 3y + 5. Divide by the bracket: x = (3y + 5)/(y - 2). Check by substituting back: with x = (3y + 5)/(y - 2), the expression (2x + 5)/(x - 3) does simplify back to y. **Example.** Now x appears twice with no fraction. Make x the subject of A = 3x + 2xy. There is no fraction to clear, so go straight to collecting: every term already contains x, so nothing moves. Factorise x out: x(3 + 2y) = A. Divide by the bracket: x = A/(3 + 2y). The same four moves work here; two of them need nothing this time. **Example.** Now x appears twice in denominators. Make x the subject of R = 1/x + 1/2. Clear the fractions by multiplying both sides by 2x: 2xR = 2 + x. Collect the x terms: 2xR - x = 2. Factorise x out: x(2R - 1) = 2. Divide: x = 2/(2R - 1). The value R = 1/2 is not allowed, because it makes the new denominator zero. **Tip.** To check a rearrangement, substitute the finished expression back into the original formula and confirm it simplifies to the other side. It takes a few lines, but it catches a dropped term or a flipped sign before you use the formula. **Recap.** Clear any fraction, collect the terms containing the new subject on one side, factorise it out, then divide by the bracket. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Simplifying Algebraic Fractions](https://lightmysky.com/learn/mathematics/simplifying-algebraic-fractions-mt_scDNSoz3eW) ## Opens up - [Completing the Square](https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm) - [Optimisation with Calculus](https://lightmysky.com/learn/mathematics/optimisation-with-calculus-mt_9I0ql39Lzw)