# 11.4 Solving equations of the form ax=b and x/a=b

 Page 1 / 2
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses solving equations of the form $ax=b$ and $\frac{x}{a}=b$ . By the end of the module students should be familiar with the multiplication/division property of equality, be able to solve equations of the form $\text{ax}=b$ and $\frac{x}{a}=b$ and be able to use combined techniques to solve equations.

## Section overview

• Multiplication/ Division Property of Equality
• Combining Techniques in Equations Solving

## Multiplication/ division property of equality

Recall that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side. From this, we can suggest the multiplication/division property of equality.

## Multiplication/division property of equality

Given any equation,

1. We can obtain an equivalent equation by multiplying both sides of the equa­tion by the same nonzero number, that is, if $c\ne 0$ , then $a=b$ is equivalent to
$a\cdot c=b\cdot c$
2. We can obtain an equivalent equation by dividing both sides of the equation by the same nonzero number , that is, if $c\ne 0$ , then $a=b$ is equivalent to
$\frac{a}{c}=\frac{b}{c}$

The multiplication/division property of equality can be used to undo an association with a number that multiplies or divides the variable.

## Sample set a

Use the multiplication / division property of equality to solve each equation.

$6y=\text{54}$
6 is associated with y by multiplication. Undo the association by dividing both sides by 6

$\begin{array}{c}\frac{6y}{6}=\frac{\text{54}}{6}\\ \frac{\overline{)6}y}{\overline{)6}}=\frac{\stackrel{9}{\overline{)54}}}{\overline{)6}}\\ y=9\end{array}$

Check: When $y=9$

$6y=\text{54}$

becomes ,
a true statement.

The solution to $6y=\text{54}$ is $y=9$ .

$\frac{x}{-2}=\text{27}$ .
-2 is associated with $x$ by division. Undo the association by multiplying both sides by -2.

$\begin{array}{}\left(-2\right)\frac{x}{-2}=\left(-2\right)\text{27}\\ \end{array}$

$\begin{array}{}\left(\overline{)-2}\right)\frac{x}{\overline{)-2}}=\left(-2\right)\text{27}\\ \end{array}$

$x=-\text{54}$

Check: When $x=-\text{54}$ ,

$\frac{x}{-2}=\text{27}$

becomes a true statement.

The solution to $\frac{x}{-2}=\text{27}$ is $x=-\text{54}$

$\frac{3a}{7}=6$ .
We will examine two methods for solving equations such as this one.

Method 1: Use of dividing out common factors.

$\frac{3a}{7}=6$
7 is associated with $a$ by division. Undo the association by multiplying both sides by 7.

$7\cdot \frac{3a}{7}=7\cdot 6$
Divide out the 7’s.

$\overline{)7}\cdot \frac{3a}{\overline{)7}}=\text{42}$

$3a=\text{42}$
3 is associated with $a$ by multiplication. Undo the association by dviding both sides by 3.

$\frac{3a}{3}=\frac{\text{42}}{3}$

$\frac{\overline{)3}a}{\overline{)3}}=\text{14}$

$a=\text{14}$

Check: When $a=\text{14}$ ,

$\frac{3a}{7}=6$

becomes ,
a true statement.

The solution to $\frac{3a}{7}=6$ is $a=\text{14}$ .

Method 2: Use of reciprocals

Recall that if the product of two numbers is 1, the numbers are reciprocals . Thus $\frac{3}{7}$ and $\frac{7}{3}$ are reciprocals.

$\frac{3a}{7}=6$
Multiply both sides of the equation by $\frac{7}{3}$ , the reciprocal of $\frac{3}{7}$ .

$\frac{7}{3}\cdot \frac{3a}{7}=\frac{7}{3}\cdot 6$

$\frac{\stackrel{1}{\overline{)7}}}{\underset{1}{\overline{)3}}}\cdot \frac{\stackrel{1}{\overline{)3}a}}{\underset{1}{\overline{)7}}}=\frac{7}{\underset{1}{\overline{)3}}}\cdot \frac{\stackrel{2}{\overline{)6}}}{1}$

$\begin{array}{c}1\cdot a=\text{14}\hfill \\ a=\text{14}\hfill \end{array}$

Notice that we get the same solution using either method.

$-8x=\text{24}$
-8 is associated with $x$ by multiplication. Undo the association by dividing both sides by -8.

$\begin{array}{c}\frac{-8x}{-8}=\frac{\text{24}}{-8}\hfill \\ \hfill \end{array}$

$\frac{-8x}{-8}=\frac{\text{24}}{-8}$

$x=-3$

Check: When $x=-3$ ,

$-8x=\text{24}$

becomes ,
a true statement.

$-x=7\text{.}$
Since $–x$ is actually $-1\cdot x$ and $\left(-1\right)\left(-1\right)=1$ , we can isolate $x$ by multiplying both sides of the equation by $-1$ .

$\begin{array}{c}\left(-1\right)\left(-x\right)=-1\cdot 7\hfill \\ x=-7\hfill \end{array}$

Check: When $x=7$ ,

$-x=7$

becomes The solution to $-x=7$ is $x=-7$ .

## Practice set a

Use the multiplication/division property of equality to solve each equation. Be sure to check each solution.

$7x=\text{21}$

$x=3$

$-5x=\text{65}$

$x=-\text{13}$

$\frac{x}{4}=-8$

$x=-\text{32}$

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
7hours 36 min - 4hours 50 min By Madison Christian By Maureen Miller By Madison Christian By Laurence Bailen By OpenStax By Yasser Ibrahim By Cameron Casey By Anh Dao By OpenStax By Edgar Delgado