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Solve: 5 ( x + 3 ) = 35 .

x = 4

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Solve: 6 ( y 4 ) = −18 .

y = 1

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Solve: ( x + 5 ) = 7 .

Solution

.
Simplify each side of the equation as much as possible by distributing.
The only x term is on the left side, so all variable terms are on the left side of the equation.
.
Add 5 to both sides to get all constant terms on the right side of the equation. .
Simplify. .
Make the coefficient of the variable term equal to 1 by multiplying both sides by -1. .
Simplify. .
Check: Let x = −12 .
.
.
.
.
.
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Solve: ( y + 8 ) = −2 .

y = −6

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Solve: ( z + 4 ) = −12 .

z = 8

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Solve: 4 ( x 2 ) + 5 = −3 .

Solution

.
Simplify each side of the equation as much as possible.
Distribute.
.
Combine like terms .
The only x is on the left side, so all variable terms are on one side of the equation.
Add 3 to both sides to get all constant terms on the other side of the equation. .
Simplify. .
Make the coefficient of the variable term equal to 1 by dividing both sides by 4. .
Simplify. .
Check: Let x = 0 .
.
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Solve: 2 ( a 4 ) + 3 = −1 .

a = 2

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Solve: 7 ( n 3 ) 8 = −15 .

n = 2

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Solve: 8 2 ( 3 y + 5 ) = 0 .

Solution

Be careful when distributing the negative.

.
Simplify—use the Distributive Property. .
Combine like terms. .
Add 2 to both sides to collect constants on the right. .
Simplify. .
Divide both sides by −6. .
Simplify. .
Check: Let y = 1 3 .
.
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Solve: 12 3 ( 4 j + 3 ) = −17 .

j = 5 3

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Solve: −6 8 ( k 2 ) = −10 .

k = 5 2

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Solve: 3 ( x 2 ) 5 = 4 ( 2 x + 1 ) + 5 .

Solution

.
Distribute. .
Combine like terms. .
Subtract 3 x to get all the variables on the right since 8 > 3 . .
Simplify. .
Subtract 9 to get the constants on the left. .
Simplify. .
Divide by 5. .
Simplify. .
Check: Substitute: −4 = x .
.
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Solve: 6 ( p 3 ) 7 = 5 ( 4 p + 3 ) 12 .

p = −2

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Solve: 8 ( q + 1 ) 5 = 3 ( 2 q 4 ) 1 .

q = −8

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Solve: 1 2 ( 6 x 2 ) = 5 x .

Solution

.
Distribute. .
Add x to get all the variables on the left. .
Simplify. .
Add 1 to get constants on the right. .
Simplify. .
Divide by 4. .
Simplify. .
Check: Let x = 3 2 .
.
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Solve: 1 3 ( 6 u + 3 ) = 7 u .

u = 2

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Solve: 2 3 ( 9 x 12 ) = 8 + 2 x .

x = 4

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In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

Solve: 0.24 ( 100 x + 5 ) = 0.4 ( 30 x + 15 ) .

Solution

.
Distribute. .
Subtract 12 x to get all the x s to the left. .
Simplify. .
Subtract 1.2 to get the constants to the right. .
Simplify. .
Divide. .
Simplify. .
Check: Let x = 0.4 .
.
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Solve: 0.55 ( 100 n + 8 ) = 0.6 ( 85 n + 14 ) .

1

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Solve: 0.15 ( 40 m 120 ) = 0.5 ( 60 m + 12 ) .

−1

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Key concepts

  • Solve an equation with variables and constants on both sides
    1. Choose one side to be the variable side and then the other will be the constant side.
    2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
    3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
    4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
    5. Check the solution by substituting into the original equation.
  • General strategy for solving linear equations
    1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
    2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
    3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
    4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.
    5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
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

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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