Basic Technical Mathematics, 11th Edition

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 ...