This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)).
Objectives of this module: be able to identify various types of equations, understand the meaning of solutions and equivalent equations, be able to solve equations of the form x + a = b and x - a = b, be familiar with and able to solve literal equations.
Overview
- Types of Equations
- Solutions and Equivalent Equations
- Literal Equations
- Solving Equations of the Form
$x+a=b$ and
$x-a=b$
Types of equations
Identity
Some equations are always true. These equations are called identities.
Identities are equations that are true for all acceptable values of the variable, that is, for all values in the domain of the equation.
$5x=5x$ is true for all acceptable values of
$x$ .
$y+1=y+1$ is true for all acceptable values of
$y$ .
$2+5=7$ is true, and no substitutions are necessary.
Contradiction
Some equations are never true. These equations are called contradictions.
Contradictions are equations that are never true regardless of the value substituted for the variable.
$x=x+1$ is never true for any acceptable value of
$x$ .
$0\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}k=14$ is never true for any acceptable value of
$k$ .
$2=1$ is never true.
Conditional equation
The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations.
Conditional equations are equations that are true for at least one replacement of the variable and false for at least one replacement of the variable.
$x+6=11$ is true only on the condition that
$x=5$ .
$y-7=-1$ is true only on the condition that
$y=6$ .
Solutions and equivalent equations
Solutions and solving an equation
The collection of values that make an equation true are called
solutions of the equation. An equation is
solved when all its solutions have been found.
Equivalent equations
Some equations have precisely the same collection of solutions. Such equations are called
equivalent equations . The equations
$\begin{array}{cccc}2x+1=7,& 2x=6& \text{and}& x=3\end{array}$
are equivalent equations because the only value that makes each one true is 3.
Sample set a
Tell why each equation is an identity, a contradiction, or conditional.
The equation
$x-4=6$ is a conditional equation since it will be true only on the condition that
$x=10$ .
The equation
$x-2=x-2$ is an identity since it is true for all values of
$x$ . For example,
$\begin{array}{cccccc}\text{if}\text{\hspace{0.17em}}x& =& 5,& 5-2& =& 5-2\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\\ x& =& -7,& -7-2& =& -7-2\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\end{array}$
The equation
$a+5=a+1$ is a contradiction since every value of
$a$ produces a false statement. For example,
$\begin{array}{cccccc}\text{if}\text{\hspace{0.17em}}a& =& 8,& 8+5& =& 8+1\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{false}\\ \text{if}\text{\hspace{0.17em}}a& =& -2,& -2+5& =& -2+1\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{false}\end{array}$
Practice set a
For each of the following equations, write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic.