The equal sign says both sides name the same amount. When an equation has a variable, it is true for some values and false for others. You can evaluate each side to find out which.
Key idea
A solution is a value of the variable that makes the equation true. To check a value, substitute it and see whether both sides are equal.
Why it works
Substituting turns the equation into a plain statement about numbers, which is either true or false. $2x + 1 = 9$ with $x = 4$ becomes $2(4) + 1 = 9$, that is $9 = 9$: true. With $x = 3$ it becomes $7 = 9$: false. The equation is a question, "which number makes this true?", and a solution is an answer to it.
Worked example: Check a value
Is $x = 4$ a solution of $2x + 1 = 9$?
- Substitute: $2(4) + 1$.
- $8 + 1 = 9$, and the right side is 9.
Answer: Yes.
Worked example: A value that fails
Is $n = 6$ a solution of $\frac{n}{3} = 3$?
- Substitute: $\frac{6}{3} = 2$.
- $2 \ne 3$.
Answer: No. (The solution is $n = 9$.)
Worked example: An inequality has many solutions
Which of 2, 3, 4 and 5 make $x + 4 > 7$ true?
- $2 + 4 = 6$: not greater than 7.
- $3 + 4 = 7$: equal, but not greater. False.
- $4 + 4 = 8 > 7$ and $5 + 4 = 9 > 7$: true.
Answer: 4 and 5
Common mistake
Two slips: stopping after working out one side, and forgetting what the sign asks for. In $x + 4 > 7$, the value 3 gives $7 > 7$, which is false: "greater than" does not include "equal to". Quick check: write the whole statement with the number in it and read it aloud. Is it true?