# 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.

$7x=\text{21}$

$x=3$

$-5x=\text{65}$

$x=-\text{13}$

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

$x=-\text{32}$

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
In the number 779,844,205 how many ten millions are there?
TELLY Reply
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
Khizar Reply
7hours 36 min - 4hours 50 min
Tanis Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

 By Steve Gibbs By Saylor Foundation By OpenStax By Jonathan Long By Richley Crapo By Jazzycazz Jackson By OpenStax By OpenStax By By Vongkol HENG