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Glacius
FoundationsConcept reference

Rearranging equations

An equivalent rearrangement preserves an equation’s solutions by applying valid inverse operations to both sides.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. An equation states that two quantities are equal
  3. Check your understanding
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01An equation states that two quantities are equal#

An equation states that two quantities are equal. To find an unknown, undo the operations around it while keeping both sides equal. In 2x+3=112x+3=11, 2x2x means two times xx.

The same operation on both sides preserves equality while the variable becomes isolated.The same operation on both sides preserves equality while the variable becomes isolated.
Figure 1The same operation on both sides preserves equality while the variable becomes isolated.
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Start with 2x+3=112x+3=11. Subtract 33 from both sides to remove the addition. Then divide both sides by 22 to leave xx alone.

2x=8x=42x=8\qquad x=4

Check the result in the original equation: 2(4)+3=112(4)+3=11. For 3x−5=73x-5=7, add 55 to both sides, then divide by 33. You again obtain x=4x=4. Always apply an operation to both entire sides.

Check your reasoning

Solve for xx: 3x+2=173x+2=17.

Show answer and explanation
5

Subtract 2 from both sides: 3x=153x=15. Divide both sides by 3: x=5x=5.

02Check your understanding#

Check your reasoning

Starting from 2x+3=92x+3=9, which step preserves equality?

  1. A2x=92x=9
  2. B2x=62x=6
  3. Cx+3=9x+3=9
Show answer and explanation
2x=62x=6

Subtract 3 from both sides, giving 2x=62x=6.

03Check your understanding#

Check your reasoning

A machine uses 22 setup minutes, then 44 minutes per item. In 1818 minutes, how many items?

Show answer and explanation
4

Remove setup time, then divide: (18−2)/4=4(18-2)/4=4.

Key takeaway

Undo operations on both sides, then check the result in the original equation.

  • Isolate one scalar variable in a linear equation

Further questions

Why check by substitution?
Putting your result into the original equation catches arithmetic and sign mistakes. Both sides must evaluate to the same number.

Sources & further reading

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Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Rearranging equations.” Math behind ML. /learn/f-rearrange