<< Chapter < Page Chapter >> Page >

Solve for a: 4 a + 2 a = 3 + 5 a 2 4 a + 2 a = 3 + 5 a 2 Add same side like terms first. 5 a + 2 = 5 a + 1 5 a + 2 5 a = 5 a + 1 5 a Subtract 5a from both sides . 10 a + 2 = 1 10 a + 2 2 = 1 2 Subtract 2 from both sides . 10 a = 1 10 a 10 = 1 10 Divide both sides by -10 . a = 1 10 The solution set is { 1 10 }

Simplifying expressions first

When solving linear equations the goal is to determine what value, if any, will solve the equation. A general guideline is to use the order of operations to simplify the expressions on both sides first.

Solve for x: 5 ( 3 x + 2 ) 2 = 2 ( 1 7 x ) 5 ( 3 x + 2 ) 2 = 2 ( 1 7 x ) Distribute . 15 x + 10 2 = 2 + 14 x Add same side like terms. 15 x + 8 = 2 + 14 x 15 x + 8 14 x = 2 + 14 x 14 x Subtract 14x on both sides. x + 8 = 2 x + 8 8 = 2 8 Subtract 8 on both sides. x = 10 The solution set is { 10 } .

Video Example 02

Conditional equations, identities, and contradictions

There are three different kinds of equations defined as follows.

Conditional Equation
A conditional equation is true for particular values of the variable.
Identity
An identity is an equation that is true for all possible values of the variable. For example, x = x has a solution set consisting of all real numbers, .
Contradiction
A contradiction is an equation that is never true and thus has no solutions. For example, x + 1 = x has no solution. No solution can be expressed as the empty set {    } = .

So far we have seen only conditional linear equations which had one value in the solution set. If when solving an equation and the end result is an identity, like say 0 = 0, then any value will solve the equation. If when solving an equation the end result is a contradiction, like say 0 = 1, then there is no solution.

Solve for x: 4 ( x + 5 ) + 6 = 2 ( 2 x + 3 ) 4 ( x + 5 ) + 6 = 2 ( 2 x + 3 ) Distribute 4 x + 20 + 6 = 4 x + 6 Add same side like terms . 4 x + 26 = 4 x + 6 4 x + 26 4 x = 4 x + 6 4 x Subtract 4x on both sides. 26 = 6 False There is no solution, .

Solve for y: 3 ( 3 y + 5 ) + 5 = 10 ( y + 2 ) y 3 ( 3 y + 5 ) + 5 = 10 ( y + 2 ) y Distribute 9 y + 15 + 5 = 10 y + 20 y Add same side like terms . 9 y + 20 = 9 y + 20 9 y + 20 20 = 9 y + 20 20 Subtract 20 on both sides . 9 y = 9 y 9 y 9 y = 9 y 9 y Subtract 9y on both sides . 0 = 0 True The equation is an identity, the solution set consists of all real numbers, .

Linear literal equations

Literal equations, or formulas, usually have more than one variable. Since the letters are placeholders for values, the steps for solving them are the same. Use the properties of equality to isolate the indicated variable.

Solve for a: P = 2 a + b P = 2 a + b P b = 2 a + b b Subtract b on both sides. P b = 2 a P b 2 = 2 a 2 Divide both sides by 2. P b 2 = a Solution: a = P b 2

Solve for x: z = x + y 2 z = x + y 2 2 z = 2 x + y 2 Multiply both sides by 2 . 2 z = x + y 2 z y = x + y y Subtract y on both sides . 2 z y = x Solution x = 2 z y

Exercises

Checking solutions

Is  x = 7  a solution to 3 x + 5 = 16 ?

Yes

Is  x = 2  a solution to  2 x 7 = 28 ?

No

Is  x = 3  a solution to  1 3 x 4 = 5 ?

Yes

Is  x = 2  a solution to  3 x 5 = 2 x 15 ?

Yes

Is  x = 1 2  a solution to  3 ( 2 x + 1 ) = 4 x 3 ?

No

Solving in one step

Solve for x:   x 5 = 8

x = 3

Solve for y:   4 + y = 9

y = 5

Solve for x:   x 1 2 = 1 3

x = 5 6

Solve for x:   x + 2 1 2 = 3 1 3

x = 5 6

Solve for x:   4 x = 44

x = 11

Solve for a:  3 a = 30

a = 10

Solve for y:   27 = 9 y

y = 3

Solve for x:   x 3 = 1 2

x = 3 2

Solve for t:   t 12 = 1 4

t = 3

Solve for x:   7 3 x = 1 2

x = 3 14

Solve in two steps

Solve for a:   3 a 7 = 23

a = 10

Solve for y:   3 y + 2 = 13

y = 5

Solve for x:   5 x + 8 = 8

x = 0

Solve for x:   1 2 x + 1 3 = 2 5

x = 2 15

Solve for y:   3 2 y = 11

y = 7

Solve for x:   10 = 2 x 5

x = 5 2

Solve for a:   4 a 2 3 = 1 6

a = 1 8

Solve for x:   3 5 x 1 2 = 1 10

x = 1

Solve for y:   4 5 y + 1 3 = 1 15

y = 1 3

Solve for x:   x 5 = 2

x = 3

Solve in multiple steps

Solve for x:   3 x 5 = 2 x 17

x = 12

Solve for y:   2 y 7 = 3 y + 13

y = 4

Solve for a:   1 2 a 2 3 = a + 1 5

a = 26 15

Solve for x:   2 + 4 x + 9 = 7 x + 8 2 x

x = 1

Solve for a:   3 a + 5 x = 2 a + 7

No Solution,

Solve for b:   7 b + 3 = 2 5 b + 1 2 b

All Reals,

Solve for y:   5 ( 2 y 3 ) + 2 = 12

y = 1 2

Solve for x:   3 2 ( x + 4 ) = 3 ( 4 x 5 )

x = 2

Solve for a:   3 ( 2 a 3 ) + 2 = 3 ( a + 7 )

a = 10 9

Solve for x:   10 ( 3 x + 5 ) 5 ( 4 x + 2 ) = 2 ( 5 x + 20 )

All Reals,

Literal equations

Solve for w:   P = 2 l + 2 w

w = P 2 l 2

Solve for b:   P = a + b + c

b = P a c

Solve for C:   F = 9 5 C + 32

C = 5 F 160 9

Solve for r:   C = 2 π r

r = C 2 π

Solve for y:   z = x y 5

y = 5 z + x

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Contemporary math applications' conversation and receive update notifications?

Ask