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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to solve equations of the form x + a = b and x - a = b . By the end of the module students should understand the meaning and function of an equation, understand what is meant by the solution to an equation and be able to solve equations of the form x + a = b size 12{x+a=b} {} and x a = b size 12{x - a=b} {} .

Section overview

  • Equations
  • Solutions and Equivalent Equations
  • Solving Equations

Equations

Equation

An equation is a statement that two algebraic expressions are equal.

The following are examples of equations:

x + 6 This expression = = 10 This expression x - 4 This expression = = - 11 This expression 3 y - 5 This expression = = - 2 + 2 y This expression

Notice that x + 6 size 12{x+6} {} , x 4 size 12{x - 4} {} , and 3 y 5 size 12{3y - 5} {} are not equations. They are expressions. They are not equations because there is no statement that each of these expressions is equal to another expression.

Solutions and equivalent equations

Conditional equations

The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations . There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.

Solutions and solving an equation

The set of values that, when substituted for the variables, make the equation true, are called the solutions of the equation.
An equation has been solved when all its solutions have been found.

Sample set a

Verify that 3 is a solution to x + 7 = 10 size 12{x+7 = "10"} {} .

When x = 3 size 12{x=3} {} ,

x + 7 = 10 becomes 3 + 7 = 10 10 = 10 which is a  true  statement, verifying that 3   is a solution to   x + 7 = 10

Verify that - 6 is a solution to 5 y + 8 = 22 size 12{5y+8= - "22"} {}

When y = - 6 size 12{y=-6} {} ,

5 y + 8 = - 22 becomes 5 ( - 6 ) + 8 = - 22 - 30 + 8 = - 22 - 22 = - 22 which is a  true  statement, verifying that - is a solution to 5 y + 8 = - 22

Verify that 5 is not a solution to a 1 = 2 a + 3 size 12{a - 1=2a+3} {} .

When a = 5 size 12{a=5} {} ,

a - 1 = 2 a + 3 becomes 5 - 1 = 2 5 + 3 5 - 1 = 10 + 3 4 = 13 a  false  statement, verifying that   5   is not a solution to  a - 1 = 2 a + 3

Verify that -2 is a solution to 3 m 2 = 4 m 16 size 12{3m - 2= - 4m - "16"} {} .

When m = - 2 size 12{x=3} {} ,

3 m - 2 = - 4 m - 16 becomes 3 ( - 2 ) - 2 = - 4 ( - 2 ) - 16 - 6 - 2 = 8 - 16 - 8 = - 8 which is a   true  statement, verifying that - 2   is a solution to  3 m - 2 = - 4 m - 16

Practice set a

Verify that 5 is a solution to m + 6 = 11 size 12{m+6="11"} {} .

Substitute 5 into m + 6 = 11 size 12{m+6="11"} {} . Does 5 plus 6 equal 11? Yes. Thus, 5 is a solution.

Verify that - 5 is a solution to 2 m 4 = 14 size 12{2m - 4= - "14"} {} .

Substitute -5 into 2 m 4 = 14 size 12{2m - 4= - "14"} {} . does 2 time negative 5 minus 4 equal negative 14? Yes. Thus, -5 is a solution.

Verify that 0 is a solution to 5 x + 1 = 1 size 12{5x+1=1} {} .

Substitute 0 into 5 x + 1 = 1 size 12{5x+1=1} {} . Does 5 times zero plus one equal 1? Yes. Thus, 0 is a solution.

Verify that 3 is not a solution to 3 y + 1 = 4 y + 5 size 12{ - 3y+1=4y+5} {} .

Substitute 3 into 3 y + 1 = 4 y + 5 size 12{ - 3y+1=4y+5} {} . Does negative 3 times 3 plus 1 equal 4 times 3 plus 5? No. Thus, 3 is not a solution.

Verify that -1 is a solution to 6 m 5 + 2 m = 7 m 6 size 12{6m - 5+2m=7m - 6} {} .

Substitute -1 into 6 m 5 + 2 m = 7 m 6 size 12{6m - 5+2m=7m - 6} {} . Does 6 times negative 1 minus 5 plus 2 times negative 1 equal 7 times negative 1 minus 6? Yes. Thus, -1 is a solution.

Equivalent equations

Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example, x - 5 = - 1 size 12{"x - 5 "=" -1"} {} , x + 7 = 11 size 12{"x "+" 7 "=" 11"} {} , and x = 4 size 12{x=4} {} are all equivalent equations since the only solution to each is x = 4 size 12{x=4} {} . (Can you verify this?)

Solving equations

We know 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.

This number is the same as this number
x size 12{x} {} = 4
x + 7 size 12{x+7} {} = 11
x 5 size 12{x - 5} {} = -1

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3
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