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How to solve literal equations (rearranging formulas)

A literal equation has several letters. Solving it means isolating one of them — the same moves as an ordinary equation, with letters instead of numbers.

阅读约 2 分钟更新于 2026年11月2日作者：Sumzo team审核：Sumzo team

A literal equation is an equation with more than one letter, such as the formula A=lwA = lw for the area of a rectangle. To solve a literal equation for one letter, you isolate it using exactly the same moves as an ordinary equation — undo each operation by doing the opposite to both sides. The answer is a formula, not a number.

Solving literal equations is also called changing the subject of a formula. It saves work: rearrange once, then substitute as many sets of numbers as you like.

How to do literal equations: the steps

  1. Circle the letter you are solving for.
  2. Undo addition and subtraction around it first, by doing the opposite to both sides.
  3. Then undo multiplication and division.
  4. If the letter appears twice, collect both terms on one side and factor it out.
  5. Check with numbers: pick values, substitute into the original and into your answer, and compare.

例题

Easy: solve A=lwA = lw for ww

Make ww the subject of A=lwA = lw.

显示解题过程
  1. ww is multiplied by ll, so divide both sides by ll.
    w=Alw = \frac{A}{l}
  2. Check with numbers: if A=24A = 24 and l=6l = 6, then 6w=246w = 24 gives w=4w = 4, and 246=4\frac{24}{6} = 4 ✓.

答案： w=Alw = \frac{A}{l}

已由 Sumzo 数学引擎核验

例题

Medium: solve y=mx+by = mx + b for xx

Rearrange the equation of a line, y=mx+by = mx + b, to make xx the subject.

显示解题过程
  1. Subtract bb from both sides.
    y−b=mxy - b = mx
  2. Divide both sides by mm.
    x=y−bmx = \frac{y - b}{m}
  3. Check: with y=11y = 11, m=2m = 2, b=3b = 3, the original gives 2x+3=112x + 3 = 11, so x=4x = 4; the formula gives 11−32=4\frac{11 - 3}{2} = 4 ✓.

答案： x=y−bmx = \frac{y - b}{m} (for m≠0m \neq 0)

已由 Sumzo 数学引擎核验

例题

Medium: the perimeter of a rectangle

Solve P=2l+2wP = 2l + 2w for ll.

显示解题过程
  1. Subtract 2w2w.
    P−2w=2lP - 2w = 2l
  2. Divide by 2.
    l=P−2w2=P2−wl = \frac{P - 2w}{2} = \frac{P}{2} - w
  3. Check: P=30P = 30, w=5w = 5 gives 2l+10=302l + 10 = 30, so l=10l = 10; and 302−5=10\frac{30}{2} - 5 = 10 ✓.

答案： l=P−2w2l = \frac{P - 2w}{2}

已由 Sumzo 数学引擎核验

例题

Hard: Celsius to Fahrenheit

Solve C=59(F−32)C = \frac{5}{9}(F - 32) for FF.

显示解题过程
  1. Multiply both sides by 95\frac{9}{5} to undo the 59\frac{5}{9}.
    95C=F−32\frac{9}{5}C = F - 32
  2. Add 32.
    F=95C+32F = \frac{9}{5}C + 32
  3. Check: water boils at C=100C = 100; the original gives 59(F−32)=100\frac{5}{9}(F - 32) = 100, so F=212F = 212, and 95(100)+32=212\frac{9}{5}(100) + 32 = 212 ✓.

答案： F=95C+32F = \frac{9}{5}C + 32

已由 Sumzo 数学引擎核验

When the letter appears twice

If the letter you want is in two terms, bring both to one side and factor it out. This is the step most students miss.

例题

Hard: factor out the variable

Solve ax+b=cx+dax + b = cx + d for xx.

显示解题过程
  1. Collect the xx terms on the left and the rest on the right.
    ax−cx=d−bax - cx = d - b
  2. Factor out xx.
    x(a−c)=d−bx(a - c) = d - b
  3. Divide by a−ca - c (which must not be 0).
    x=d−ba−cx = \frac{d - b}{a - c}
  4. Check: 5x+2=3x+105x + 2 = 3x + 10 has x=4x = 4, and 10−25−3=4\frac{10 - 2}{5 - 3} = 4 ✓.

答案： x=d−ba−cx = \frac{d - b}{a - c}, for a≠ca \neq c

已由 Sumzo 数学引擎核验

Common mistakes when solving literal equations

  • Dividing only one term: from P−2w=2lP - 2w = 2l, divide the whole left side by 2.
  • Cancelling letters that are added, not multiplied: y−bm\frac{y - b}{m} is not y−bmy - \frac{b}{m}.
  • Leaving the variable on both sides — factor it out.
  • Forgetting the conditions: you can’t divide by mm if m=0m = 0.

常见问题

What is a literal equation?

An equation with two or more letters, usually a formula such as d=rtd = rt or A=πr2A = \pi r^2.

How do you solve literal equations step by step?

Undo the operations around the letter you want in reverse order — addition and subtraction first, then multiplication and division — doing the same to both sides. Factor if the letter appears twice.

How can I check my answer?

Choose numbers for the other letters, solve the original equation with them, and see if your rearranged formula gives the same value.

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