You had \pounds 12 in your pocket, bought a snack, and now have
\pounds 7 left — so how much did the snack cost? Whenever you know
a total and every piece but one, finding that missing piece is what
solving an equation means, and it turns up with money, distances, scores and
recipes all the time.
An equation says two things are equal. The equals sign is a
balance: whatever sits on the left weighs exactly the same as whatever sits on
the right. Solving for x means getting it alone on one side — without
ever tipping that balance.
Picture a pair of kitchen scales. The two pans hang level only because both sides hold the same
weight. If you quietly took a marble off just one pan, that side would shoot up and the scales
would tell a lie. So there is one golden rule: whatever you do to one side, do to the
other. Keep them equal and the scales stay honest.
The way to free x is the inverse operation: to undo
something, do its opposite. Adding is undone by subtracting; multiplying is undone by dividing.
Take this equation:
x + 5 = 12
Here 5 was added to x, so we undo
it by subtracting 5 from both sides. The left becomes
x on its own; the right becomes 7:
x + 5 - 5 = 12 - 5 \quad\Longrightarrow\quad x = 7
Imagine a see-saw that is perfectly level because two friends of equal weight sit on the ends.
If one friend hops off, that end flies up. But if both friends hop off at the same
moment, the see-saw stays level. An equation is the same: removing 5 from only one side breaks
the balance, but removing 5 from both sides keeps everything equal — and now
x is sitting all by itself.
Two worked examples
Example 1 — undo an addition. Solve x + 8 = 13. The
8 is added to x, so subtract
8 from both sides:
x + 8 - 8 = 13 - 8 \quad\Longrightarrow\quad x = 5
Check by substituting back: put x = 5 into the
original equation. Does 5 + 8 = 13? Yes! So
x = 5 is correct.
Example 2 — undo a multiplication. Multiplication works the same way with its
inverse, division. In 3x = 12, the x is
multiplied by 3, so divide both sides by
3:
\frac{3x}{3} = \frac{12}{3} \quad\Longrightarrow\quad x = 4
Check: does 3 \times 4 = 12? Yes — so
x = 4. Substituting your answer back into the start is the surest way
to know you have it right; if both sides come out equal, you have solved it. The inverses
themselves are just
division and
subtraction
used to peel away whatever surrounds the unknown, and
substituting a value back in
is how you check.
Two rules keep one-step equations honest:
- Whatever you do to one side, do to the other. Subtract 5 from the left?
Subtract 5 from the right too. Touch only one pan and the scales tip — your answer will be
wrong.
- Undo with the INVERSE operation. To free an added number you
subtract; to free a multiplied number you divide. Doing the same
operation again (adding when you meant to subtract) just makes things worse, not simpler.