Math Vale

⚔️ Algebra Kingdom · Algebra

Literal Equations & Formulas

Rearrange a formula to isolate any one of its letters, using the same moves as a numeric equation, and use the rearranged form to solve for the quantity you actually need.

In short

  • A literal equation is a rule linking several letters, and solving it for one letter means rewriting the rule with that letter alone on one side.
  • Choose the letter you want and treat every other letter exactly as you would treat a number you happen not to know.
  • The moves are the ones you already use: distribute, combine, collect the letter you want on one side, then undo — outermost operation first.
  • When you divide, the whole of the other side is divided, which is why the answer so often needs a bracket around it.
  • A formula with letters left in it is a finished answer, because nothing can simplify until real numbers arrive.
  • Rearrange while the formula is still letters and substitute the numbers once at the end.

A formula is an equation wearing more letters

2x + 7 = 19 has one letter, and solving it means finding the one number x can be.

p = 2l + 2w has three, and it is not asking you to find anything. It is a rule: whatever the length and width of a rectangle happen to be, this is how its perimeter is built. An equation with more than one letter is called a literal equation, and rearranging one is called *solving for* a letter.

Here is the idea that makes the whole topic easy. Pick the letter you want and treat every other letter as if it were a number. In p = 2l + 2w, if you are solving for w, then p and l are just numbers whose names you happen not to know. You would solve p = 2l + 2w exactly the way you would solve 96 = 60 + 2w.

Two things change, and only two:

  • Nothing tidies itself up at the end. 96 - 60 collapses to 36, but p - 2l stays as p - 2l. That is not an unfinished answer; it *is* the answer.
  • You never find out what the letter equals as a number. You end up with a new formula, and a formula is a perfectly good destination.

The same four moves, in the same order

Rearranging a formula uses the moves you already know from multi-step equations, in the order you already know: distribute, combine, collect the letter you want on one side, undo.

To undo, work from the outside in — peel off whatever was done to your letter last, first. Solving p = 2l + 2w for w:

p = 2l + 2w p - 2l = 2w (subtract 2l from both sides) (p - 2l)/2 = w (divide both sides by 2)

The second move is where most marks are lost. When you divide, the whole of the other side is divided, every term of it — which is exactly what the bracket in (p - 2l)/2 is for. Writing p/2 - 2l instead halves the p and leaves the 2l untouched, and that is a different formula.

Any equivalent form is right. (p - 2l)/2 and p/2 - l are the same rule written two ways; pick whichever you find easier to read, and check by expanding.

Whatever you do, do it to both sides. That rule has not changed since the first equation you ever met, and it is the only permission you need for any move here.

When the letter is multiplied, divided or squared

Not every letter is added on. Ask two questions about the letter you want: what is multiplying it, and what is dividing it?

Multiplied by letters. d = rt, solved for t: t is multiplied by r, so divide both sides by r, giving t = d/r. In v = lwh, h is multiplied by *both* l and w, so both have to go: h = v/(lw). Dividing by only one of them leaves the other stuck to h.

Sitting underneath. In r = d/t, the t you want is below the line. Multiply both sides by t first to lift it up: rt = d, and then t = d/r.

With a number attached. a = bh/2 has a 2 underneath. Clear it before anything else: multiply both sides by 2 to get 2a = bh, then divide by b, giving h = 2a/b. Leaving the 2 behind is the commonest slip in this whole topic.

Squared or rooted. In a = s2, the last thing done to s was squaring, so undo it last: s = sqrt(a). If something else is attached — a = 5s2 — clear that first, so s2 = a/5 and then s = sqrt(a/5). Order matters: sqrt(a)/5 and sqrt(a/5) are different numbers.

Rearrange first, substitute second

Suppose c = 5(f - 32)/9 and you know that c = 25. What is f?

You could put 25 in straight away and unpick the equation that results. That works, but you have to do it again for every new temperature. Rearranging first gives you a tool you can use for ever:

c = 5(f - 32)/9 9c = 5(f - 32) (multiply both sides by 9) 9c/5 = f - 32 (divide both sides by 5) 9c/5 + 32 = f (add 32 to both sides)

Now substitute: f = 9 x 25/5 + 32 = 45 + 32 = 77.

The rule of thumb: rearrange while the formula is still letters, then put the numbers in once. With letters there is only one copy of the thing you want to chase. With numbers in the way, you are back to solving an equation from scratch every single time.

The same habit answers word problems. Name the formula the story is built on, decide which letter the story is asking for, rearrange for that letter, and only then substitute.

Why this matters later

Rearranging formulas is the quiet skill behind a great deal of the mathematics ahead of you.

  • Science lives on it. d = rt, f = ma, i = prt and v = lwh are all asked in every direction. A physicist who could only ever solve for the letter on the left would be stuck almost immediately.
  • Straight lines. y = mx + b solved for x, or for m, is how you read a gradient off a graph and how you turn one form of a line's equation into another.
  • Functions and inverses. Swapping y = 2x + 3 into x = (y - 3)/2 is exactly the rearranging in this lesson, and it is how an inverse function is found.
  • Coefficients that are letters. ax + b = c is the general shape of *every* two-step equation, and solving it once, as x = (c - b)/a, solves all of them at a stroke. That is what a formula is for: doing a whole family of calculations in advance.

When you meet the quadratic formula later, it will look intimidating. It is nothing more than ax2 + bx + c = 0 solved for x, by someone who did the rearranging once so that nobody else ever has to.

Worked examples

Example 1

Solve p = 2l + 2w for l.

  1. Only l has to end up alone. Treat p and w as numbers whose names you do not know.
  2. Subtract 2w from both sides: p - 2w = 2l.
  3. Divide both sides by 2 — the whole of the left side, not just the p: (p - 2w)/2 = l.
  4. So l = (p - 2w)/2, which can also be written l = p/2 - w. Nothing else cancels, so that is finished.

Example 2

Solve a = bh/2 for h.

  1. h is multiplied by b and divided by 2, so both have to be undone.
  2. Clear the bottom line first: multiply both sides by 2, giving 2a = bh.
  3. Now h is only multiplied by b, so divide both sides by b: 2a/b = h.
  4. So h = 2a/b. Check with numbers: if a = 24 and b = 6, then h = 48/6 = 8, and indeed 6 x 8/2 = 24.

Example 3

Solve ax + b = cx + d for x.

  1. There is an x on each side, so nothing can be divided yet — collect the x terms first.
  2. Subtract cx and b from both sides: ax - cx = d - b.
  3. The left side has a common factor of x, so factor it out: (a - c)x = d - b.
  4. Divide both sides by the whole coefficient a - c: x = (d - b)/(a - c).

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Solve for g: z = g + 2d Give g as an expression in the other letters.

Answer: z - 2*d

  1. z = g + 2d
  2. g = z - 2d (subtract 2d from both sides)
  3. Every other letter stands for a fixed but unknown number, so nothing cancels at the end — a formula IS the answer.

Problem 2

Difficulty 3 of 5

This formula gives the volume v of a box l long, w wide and h high: v = lwh Rearrange it to solve for l. Give l as an expression in the other letters.

Answer: v/(w*h)

  1. v = lwh
  2. l = v/(wh) (divide both sides by wh)
  3. Every other letter stands for a fixed but unknown number, so nothing cancels at the end — a formula IS the answer.

Problem 3

Difficulty 4 of 5

This formula gives the pull f between two lodestones of strength g and m held a distance d apart: f = gm/d Which rearrangement solves it for d?

  1. m = fd/g
  2. d = f/(gm)
  3. d = f - f
  4. d = gm/f

Answer: D. d = gm/f

  1. f = gm/d
  2. fd = gm (multiply both sides by d)
  3. d = gm/f (divide both sides by f)
  4. So the correct rearrangement is d = gm/f.

Common mistakes

  • Moving a term without changing its sign: turning p = 2l + 2w into 2w = p + 2l. A term that crosses the equals sign reverses.
  • Dividing only one term by the coefficient: writing w = p/2 - 2l instead of w = (p - 2l)/2. The divisor shares out every term on that side.
  • Subtracting instead of dividing: reading d = rt as t = d - r. The r is multiplying t, so only division sets t free.
  • Solving for the wrong letter: rearranging p = 2l + 2w for l when the question asked for w. Read the letter off the question before you start.
  • Leaving a number behind: turning a = bh/2 into h = a/b and forgetting that the 2 has to be cleared first.
  • Expecting the answer to simplify to a single number, and assuming something has gone wrong when it does not. A formula is the answer here.
  • Taking a root before clearing what is beside it: reading a = 5s2 as s = sqrt(a)/5 rather than s = sqrt(a/5).

What you should be able to do

  • Solve a formula such as p = 2l + 2w for one of its variables.
  • Isolate a variable that appears inside a fraction or multiplied by another letter.
  • Choose the rearrangement that matches a given formula.
  • Rearrange a formula and then evaluate it for given values.

Where this fits in the curriculum

Common Core

  • HSA-CED.A.4

    High school — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

  • HSA-REI.B.3

    High school — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Ontario

  • MTH1W.C1.5

    Grade 9 de-streamed — Create and solve equations for various contexts, and verify their solutions.

    Ontario does not name rearranging a formula for a chosen variable before the senior courses; the Grade 9 equation-solving expectation is the nearest home, so this pairing is a judgement call.

SAT

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