What is Equation, solution? — Mathematics

Definition 2.7 High School Mathematics · Chapter 2 — Algebra: Equations and Inequalities

An equation is an equality containing an unknown number $x$ ; a solution is a value of $x$ that makes the equality true. Solving the equation means finding all its solutions.

Examples

Example 2.8 (First-degree equation)

Solve $5x - 7 = 2x + 8$ , one move at a time:

\begin{align*} 5x - 7 &= 2x + 8 \\ 5x - 2x &= 8 + 7 && \text{(add $7 - 2x$ to both sides)} \\ 3x &= 15 \\ x &= 5 && \text{(divide both sides by $3$).} \end{align*}

The unique solution is $5$ . Check: $5 \times 5 - 7 = 18$ and $2 \times 5 + 8 = 18$ .

Example 2.11

Solve $(x - 2)(3x + 6) = 0$ : either $x - 2 = 0$ , giving $x = 2$ , or $3x + 6 = 0$ , giving $x = -2$ . Solutions: $-2$ and $2$ .

Solve $x^2 = 9x$ . Do not divide by $x$ (it could be zero!); bring everything to one side and factor:

x^2 - 9x = 0 \quad\Longleftrightarrow\quad x(x - 9) = 0 \quad\Longleftrightarrow\quad x = 0 \ \text{ or } \ x = 9 .

Example 2.12 (Quotient equation)

Solve $\dfrac{x - 1}{x + 2} = 0$ . A quotient is zero exactly when its numerator is zero and its denominator is not. Here $x - 1 = 0$ gives $x = 1$ , and $1 + 2 = 3 \neq 0$ : the only solution is $1$ . The value $x = -2$ is forbidden (division by zero) and must always be excluded first.

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