# Elementary algebra: solving linear equations in one variable

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Elementary Algebra: An introduction to solving linear equations in one variable.

## Module overview

Learning how to solve various algebraic equations is one of our main goals in algebra. This module introduces the basic techniques for solving linear equations in one variable. (Prerequisites: Working knowledge of real numbers and their operations.)

## Objectives

• Define Linear Equations in One Variable
• Solutions to Linear Equations
• Solving Linear Equations
• Combining Like Terms and Simplifying
• Literal Equations

## Define linear equations in one variable

We begin by establishing some definitions.

Equation
An equation is a statement indicating that two algebraic expressions are equal.
Linear Equation in One Variable
A linear equation in one variable $x$ is an equation that can be written in the form $\text{ax}+b=0$ where $a$ and $b$ are real numbers and $a\ne 0$ .

Following are some examples of linear equations in one variable, all of which will be solved in the course of this module.

$x+3=-5$
$\frac{x}{3}+\frac{1}{2}=\frac{2}{3}$
$5\left(3x+2\right)-2=-2\left(1-7x\right)$

## Solutions to linear equations in one variable

The variable in the linear equation $2x+3=\text{13}$ is $x$ . Values that can replace the variable to make a true statement compose the solution set. Linear equations have at most one solution. After some thought, you might deduce that $x=5$ is a solution to $2x+3=\text{13}$ . To verify this we substitute the value 5 in for $x$ and see that we get a true statement, $2\left(\mathbf{5}\right)+3=\text{10}+3=\text{13}$ .

Is $x=3$ a solution to $-2x-3=-9$ ?

Yes, because $-2\left({3}\right)-3=-6-3=-9$

Is $a=-\frac{1}{2}$ a solution to $-\text{10}a+5=\text{25}$ ?

No, because $-\text{10}\left({-}\frac{{1}}{{2}}\right)+5=5+5=\text{10}\ne \text{25}$

When evaluating expressions, it is a good practice to replace all variables with parenthesis first, then substitute in the appropriate values. By making use of parenthesis we could avoid some common errors using the order of operations.

Is $y=-3$   a solution to $2y-5=-y-\text{14}$ ? Yes because $y=-3$ produces a true mathematical statement.

## Solving linear equations in one variable

When the coefficients of linear equations are numbers other than nice easy integers, guessing at solutions becomes an unreasonable prospect. We begin to develop an algebraic technique for solving by first looking at the properties of equality.

## Properties of equality

Given algebraic expressions A and B where c is a real number:

## Division property of equality

Multiplying or dividing both sides of an equation by zero is carefully avoided. Dividing by zero is undefined and multiplying both sides by zero will result in an equation 0=0.

To summarize, the equality is retained if we add, subtract, multiply and divide both sides of an equation by any nonzero real number. The central technique for solving linear equations involves applying these properties in order to isolate the variable on one side of the equation.

Use the properties of equality to solve: $x+3=-5$ The solution set is $\left\{-8\right\}$ .

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Ali
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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
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da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
How we are making nano material?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
what is differents between GO and RGO?
Mahi
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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
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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
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