# 5.3 Further techniques in equation solving

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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 comfortable with combining techniques in equation solving, be able to recognize identities and contradictions.

## Overview

• Combining Techniques in Equation Solving
• Recognizing Identities and Contrdictions

## Combining techniques in equation solving

In Sections [link] and [link] we worked with techniques that involved the use of addition, subtraction, multiplication, and division to solve equations. We can combine these techniques to solve more complicated equations. To do so, it is helpful to recall that an equation is solved for a particular variable when all other numbers and/or letters have been disassociated from it and it is alone on one side of the equal sign. We will also note that

To associate numbers and letters we use the order of operations.

1. Multiply/divide

To undo an association between numbers and letters we use the order of operations in reverse.

2. Multiply/divide

## Sample set a

Solve $4x-7=9$ for $x.$

$\begin{array}{llll}\hfill 4x-7& =\hfill & 9\hfill & \text{First,}\text{\hspace{0.17em}}\text{undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{between}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}7.\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{adding}\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{.}\hfill \\ \hfill 4x-7+7& =\hfill & 9+7\hfill & \hfill \\ \hfill 4x& =\hfill & 16\hfill & \text{Now,}\text{\hspace{0.17em}}\text{undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{between}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}4.\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}.\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill \frac{4x}{4}& =& \frac{16}{4}& \\ \hfill 16-7& =\hfill & 9\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill x& =\hfill & 4\hfill & \hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 4\left(4\right)-7& =\hfill & 9\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 9& =\hfill & 9\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Solve $\frac{3y}{4}-5=-11.$

$\begin{array}{llll}\hfill \frac{3y}{4}-5& =\hfill & -11\hfill & -5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{adding}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{.}\hfill \\ \hfill \frac{3y}{4}-5+5& =\hfill & -11+5\hfill & \hfill \\ \hfill \frac{3y}{4}& =\hfill & -6\hfill & 4\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{division}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplying}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill 4\cdot \frac{3y}{4}& =\hfill & 4\left(-6\right)\hfill & \hfill \\ \hfill 4\cdot \frac{3y}{4}& =\hfill & 4\left(-6\right)\hfill & \hfill \\ \hfill 3y& =\hfill & -24\hfill & 3\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}3.\hfill \\ \hfill \frac{3y}{3}& =& \frac{-24}{3}& \hfill \\ \hfill \frac{3y}{3}& =\hfill & -8\hfill & \hfill \\ \hfill y& =\hfill & -8\hfill & \hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill \frac{3\left(-8\right)}{4}-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill \frac{-24}{4}-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -6-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -11& =\hfill & -11\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Solve $\frac{8a}{3b}+2m=6m-5$ for $a.$

$\begin{array}{llll}\hfill \frac{8a}{3b}+2m& =\hfill & 6m-5\hfill & 2m\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{addition}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtracting}\text{\hspace{0.17em}}2m\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill \frac{8a}{3b}+2m-2m& =\hfill & 6m-5-2m\hfill & \hfill \\ \hfill \frac{8a}{3b}& =\hfill & 4m-5\hfill & 3b\text{\hspace{0.17em}}\text{\hspace{0.17em}}associated\text{\hspace{0.17em}}with\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{division}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplying}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}3b.\hfill \\ \hfill \left(3b\right)\left(\frac{8a}{3b}\right)& =\hfill & 3b\left(4m-5\right)\hfill & \hfill \\ \hfill 8a& =\hfill & 12bm-15b\hfill & 8\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\hfill \\ \hfill & \hfill & \hfill & \text{multiplication}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}8.\hfill \\ \hfill \frac{8a}{8}& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \\ \hfill a& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \end{array}$

## Practice set a

Solve $3y-1=11$ for $y.$

$y=4$

Solve $\frac{5m}{2}+6=1$ for $m.$

$m=-2$

Solve $2n+3m=4$ for $n.$

$n=\frac{4-3m}{2}$

Solve $\frac{9k}{2h}+5=p-2$ for $k.$

$k=\frac{2hp-14h}{9}$

Sometimes when solving an equation it is necessary to simplify the expressions composing it.

## Sample set b

Solve $4x+1-3x=\left(-2\right)\left(4\right)$ for $x.$

$\begin{array}{lll}\hfill 4x+1-3x& =\hfill & \left(-2\right)\left(4\right)\hfill \\ \hfill x+1& =\hfill & -8\hfill \\ \hfill x& =\hfill & -9\hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 4\left(-9\right)+1-3\left(-9\right)& =\hfill & -8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -36+1+27& =\hfill & -8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -8& =\hfill & -8\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

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