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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 size 12{x} {} is an equation that can be written in the form ax + b = 0 size 12{ ital "ax"+b=0} {} where a size 12{a} {} and b size 12{b} {} are real numbers and a 0 size 12{a<>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
x 3 + 1 2 = 2 3
5 ( 3 x + 2 ) 2 = 2 ( 1 7 x )

Solutions to linear equations in one variable

The variable in the linear equation 2x + 3 = 13 size 12{2x+3="13"} {} is x size 12{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 size 12{x=5} {} is a solution to 2x + 3 = 13 size 12{2x+3="13"} {} . To verify this we substitute the value 5 in for x size 12{x} {} and see that we get a true statement, 2 5 + 3 = 10 + 3 = 13 size 12{2 left (5 right )+3="10"+3="13"} {} .

Is x = 3 size 12{x=3} {} a solution to 2x 3 = 9 size 12{ - 2x - 3= - 9} {} ?

Yes, because 2 3 3 = 6 3 = 9 size 12{ - 2 left (3 right ) - 3= - 6 - 3= - 9} {}

Is a = 1 2 size 12{a= - { { size 8{1} } over { size 8{2} } } } {} a solution to 10 a + 5 = 25 size 12{ - "10"a+5="25"} {} ?

No, because 10 1 2 + 5 = 5 + 5 = 10 25 size 12{ - "10" left ( - { { size 8{1} } over { size 8{2} } } right )+5=5+5="10"<>"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 size 12{y= - 3} {}   a solution to 2y 5 = y 14 size 12{2y - 5= - y - "14"} {} ? 2 (    ) 5 = (    ) 14 Replace variables with parenthesis . 2 ( 3 ) 5 = ( 3 ) 14 Substitute the appropriate value . 6 5 = 3 14 Simplify . 11 = 11 True . 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:

Addition property of equality

If  A = B  then  A + c = B + c

Subtraction property of equality

If  A = B  then  A - c = B - c

Multiplication property of equality

If  A = B  and  c 0  then c A = c B

Division property of equality

If  A = B  and  c 0  then  A c = B c

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 x + 3 = 5 x + 3 3 = 5 3 Subtract 3 on both sides . x = 8 Simplify The solution set is { 8 } .

Questions & Answers

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
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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