# 1.10 Solving Equations

**Types of Equations • Solving Basic Types of Equations • Checking the Solution • First Steps • Ratio and Proportion**

In this section, we show how algebraic operations are used in solving equations. In the following sections, we show some of the important applications of equations.

*An* **equation** *is an algebraic statement that two algebraic expressions are equal*. Any value of the **unknown** that produces equality when **substituted** in the equation is said to **satisfy** the equation and is called a **solution** of the equation.

# EXAMPLE 1 Valid values for equations

The equation $3x\hspace{0.17em}-\hspace{0.17em}5\hspace{0.17em}=\hspace{0.17em}x\hspace{0.17em}+\hspace{0.17em}1$ is true only if $x\hspace{0.17em}=\hspace{0.17em}3.$ Substituting 3 for `x` in the equation, we have $3(3)\hspace{0.17em}-\hspace{0.17em}5\hspace{0.17em}=\hspace{0.17em}3\hspace{0.17em}+\hspace{0.17em}1\hspace{0.17em},\hspace{0.17em}$ or $4\hspace{0.17em}=\hspace{0.17em}4\hspace{0.17em};\hspace{0.17em}$ substituting $x\hspace{0.17em}=\hspace{0.17em}2\hspace{0.17em},\hspace{0.17em}$ we have $1\hspace{0.17em}=\hspace{0.17em}3\hspace{0.17em},\hspace{0.17em}$ which ...

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