Equations: what they are and how to solve them

An equation is a mathematical statement that two expressions are equal. The equals sign sits in the middle, with one expression on each side: $3x + 5 = 20$ says “three times a number, plus five, is twenty”. The letter (here $x$) stands for a number we don’t know yet.

Solving an equation means finding the value (or values) of the unknown that make both sides equal. That value is the solution. For $3x + 5 = 20$ the solution is $x = 5$, because $3 \times 5 + 5 = 20$. Every math equation in this lesson is solved that way: find the number, then check it.

What is the solution to an equation?

The solution is the number that makes the equation true when you substitute it for the letter. Some equations have one solution, some have two (like $x^2 = 9$, solved by $3$ and $-3$), some have none (like $x + 1 = x + 2$) and some are true for every number (like $2(x + 1) = 2x + 2$).

Types of equations you will meet

• Linear equations: the unknown appears only to the first power, like $5x - 3 = 2x + 9$. One solution (or none, or every number).
• Quadratic equations: the unknown is squared, like $x^2 - 5x + 6 = 0$. Usually two solutions.
• Simultaneous equations (systems): two or more equations with two or more unknowns that must all be true at once.
• Literal equations (formulas): several letters, like $v = u + at$. Solving for one letter rearranges the formula: $t = \frac{v - u}{a}$.

How to solve equations: keep them balanced

Think of an equation as a balance. Whatever you do to one side, do exactly the same to the other, and it stays balanced. To get the unknown on its own, undo what has been done to it, in reverse order.

1. Expand any brackets and simplify each side.
2. Move the terms with the unknown to one side and the numbers to the other, by adding or subtracting on both sides.
3. Divide both sides by the number in front of the unknown.
4. Check: substitute your answer into the original equation.

How to solve equations with two variables

One equation with two unknowns, like $x + y = 10$, has endless solutions ($1$ and $9$, $2$ and $8$, …). To pin down one answer you need a second equation. Together they form a system of equations, solved by elimination or substitution.

When the unknown is squared

Balancing alone can’t isolate $x$ when it appears as $x^2$ and as $x$. Move everything to one side so the equation equals zero, factor it, and use the fact that a product is zero only when one of its factors is zero.

Turning words into an equation

Many math equation questions start as a sentence. Give the unknown a letter, then translate piece by piece: “is” becomes $=$, “more than” or “increased by” becomes $+$, “times” or “doubled” becomes multiplication.

Complete the equation (missing number problems)

Younger students often meet equations as a box to fill in. The box is just an unknown, so the same idea works: undo the operation.

• 3 × ☐ = 24: divide 24 by 3, so the box is 8.
• ☐ − 15 = 27: add 15 to 27, so the box is 42.
• 56 ÷ ☐ = 7: 56 divided by what is 7? The box is 8, because 56 ÷ 8 = 7.
• 2 × ☐ + 5 = 17: subtract 5 to get 12, then divide by 2, so the box is 6.

Easy math equations with answers to try

These simple math equations use the same steps as the examples. Cover the answers, solve each one, then check by substituting.

1. $x - 9 = 4$ → $x = 13$
2. $6x = 42$ → $x = 7$
3. $3x + 2 = 20$ → $x = 6$
4. $7 - 2x = 1$ → $x = 3$
5. $2(x + 4) = 18$ → $x = 5$

Common mistakes

• Doing something to one side only. If you subtract 3 on the left, subtract 3 on the right too.
• Sign slips when moving terms: from $5x - 3 = 2x + 9$, subtracting $2x$ gives $3x - 3 = 9$, not $7x - 3 = 9$.
• Dividing only part of a side: from $2x = 6 + 4$, dividing just the 6 gives $x = 3 + 4 = 7$, which is wrong. Divide every term ($3 + 2$) or simplify first ($2x = 10$): $x = 5$.
• Not multiplying every term when clearing fractions — in the fractions example, both whole brackets are multiplied, giving $3x + 9$ and $8x - 4$.
• Skipping the check. Substituting takes seconds and catches most errors.