# 5.1 Solving linear equations

 Page 1 / 1

## Equations and inequalities: solving linear equations

The simplest equation to solve is a linear equation. A linear equation is an equation where the power of the variable(letter, e.g. $x$ ) is 1(one). The following are examples of linear equations.

$\begin{array}{ccc}\hfill 2x+2& =& 1\hfill \\ \hfill \frac{2-x}{3x+1}& =& 2\hfill \\ \hfill \frac{4}{3}x-6& =& 7x+2\hfill \end{array}$

In this section, we will learn how to find the value of the variable that makes both sides of the linear equation true. For example, what value of $x$ makes both sides of the very simple equation, $x+1=1$ true.

Since the definition of a linear equation is that if the variable has a highest power of one (1), there is at most one solution or root for the equation.

This section relies on all the methods we have already discussed: multiplying out expressions, grouping terms and factorisation. Make sure that you arecomfortable with these methods, before trying out the work in the rest of this chapter.

$\begin{array}{cccc}\hfill 2x+2& =& 1\hfill & \\ \hfill 2x& =& 1-2\hfill & \left(\mathrm{like terms together}\right)\hfill & \\ \hfill 2x& =& -1\hfill & \left(\mathrm{simplified as much as possible}\right)\hfill & \end{array}$

Now we see that $2x=-1$ . This means if we divide both sides by 2, we will get:

$x=-\frac{1}{2}$

If we substitute $x=-\frac{1}{2}$ , back into the original equation, we get:

$\begin{array}{ccc}\hfill \mathrm{LHS}& =& 2x+2\hfill \\ & =& 2\left(-\frac{1}{2}\right)+2\hfill \\ & =& -1+2\hfill \\ & =& 1\hfill \\ \hfill \mathrm{and}\\ \hfill \mathrm{RHS}& =& 1\hfill \end{array}$

That is all that there is to solving linear equations.

## Solving equations

When you have found the solution to an equation, substitute the solution into the original equation, to check your answer.

## Method: solving linear equations

The general steps to solve linear equations are:

1. Expand (Remove) all brackets that are in the equation.
2. "Move" all terms with the variable to the left hand side of the equation, and all constant terms (the numbers) to the right hand side of the equals sign.Bearing in mind that the sign of the terms will change from ( $+$ ) to ( $-$ ) or vice versa, as they "cross over" the equals sign.
3. Group all like terms together and simplify as much as possible.
4. If necessary factorise.
5. Find the solution and write down the answer(s).
6. Substitute solution into original equation to check answer.

Solve for $x$ : $4-x=4$

1. We are given $4-x=4$ and are required to solve for $x$ .

2. Since there are no brackets, we can start with rearranging and then grouping like terms.

3. $\begin{array}{cccc}\hfill 4-x& =& 4\hfill & \\ \hfill -x& =& 4-4\hfill & \left(\mathrm{Rearrange}\right)\hfill \\ \hfill -x& =& 0\hfill & \left(\mathrm{group like terms}\right)\hfill \\ \hfill âˆ´\phantom{\rule{1.em}{0ex}}x& =& 0\hfill & \end{array}$
4. Substitute solution into original equation:

$\begin{array}{ccc}\hfill 4-0& =& 4\hfill \\ \hfill 4& =& 4\hfill \end{array}$

Since both sides are equal, the answer is correct.

5. The solution of $4-x=4$ is $x=0$ .

Solve for $x$ : $4\left(2x-9\right)-4x=4-6x$

1. We are given $4\left(2x-9\right)-4x=4-6x$ and are required to solve for $x$ .

2. We start with expanding the brackets, then rearranging, then grouping like terms and then simplifying.

3. $\begin{array}{cccc}\hfill 4\left(2x-9\right)-4x& =& 4-6x\hfill & \\ \hfill 8x-36-4x& =& 4-6x\hfill & \left(\mathrm{expand the brackets}\right)\hfill \\ \hfill 8x-4x+6x& =& 4+36\hfill & \left(\mathrm{Rearrange}\right)\hfill \\ \hfill \left(8x-4x+6x\right)& =& \left(4+36\right)\hfill & \left(\mathrm{group like terms}\right)\hfill \\ \hfill 10x& =& 40\hfill & \left(\mathrm{simplify grouped terms}\right)\hfill \\ \hfill \frac{10}{10}x& =& \frac{40}{10}\hfill & \left(\mathrm{divide both sides by}\phantom{\rule{2pt}{0ex}}10\right)\hfill \\ \hfill x& =& 4\hfill & \end{array}$
4. Substitute solution into original equation:

$\begin{array}{ccc}\hfill 4\left(2\left(4\right)-9\right)-4\left(4\right)& =& 4-6\left(4\right)\hfill \\ \hfill 4\left(8-9\right)-16& =& 4-24\hfill \\ \hfill 4\left(-1\right)-16& =& -20\hfill \\ \hfill -4-16& =& -20\hfill \\ \hfill -20& =& -20\hfill \end{array}$

Since both sides are equal to $-20$ , the answer is correct.

5. The solution of $4\left(2x-9\right)-4x=4-6x$ is $x=4$ .

Solve for $x$ : $\frac{2-x}{3x+1}=2$

1. We are given $\frac{2-x}{3x+1}=2$ and are required to solve for $x$ .

2. Since there is a denominator of ( $3x+1$ ), we can start by multiplying both sides of the equation by ( $3x+1$ ). But because division by 0 is not permissible, there is a restriction on a value for x. ( )

3. $\begin{array}{cccc}\hfill \frac{2-x}{3x+1}& =& 2\hfill & \\ \hfill \left(2-x\right)& =& 2\left(3x+1\right)\hfill & \\ \hfill 2-x& =& 6x+2\hfill & \left(\mathrm{expand brackets}\right)\hfill \\ \hfill -x-6x& =& 2-2\hfill & \left(\mathrm{rearrange}\right)\hfill \\ \hfill -7x& =& 0\hfill & \left(\mathrm{simplify grouped terms}\right)\hfill \\ \hfill x& =& 0Ã·\left(-7\right)\hfill & \\ \hfill âˆ´\phantom{\rule{2.em}{0ex}}x& =& 0\hfill & \left(\mathrm{zero divided by any number is}\phantom{\rule{3pt}{0ex}}0\right)\hfill \end{array}$
4. Substitute solution into original equation:

$\begin{array}{ccc}\hfill \frac{2-\left(0\right)}{3\left(0\right)+1}& =& 2\hfill \\ \hfill \frac{2}{1}& =& 2\hfill \end{array}$

Since both sides are equal to 2, the answer is correct.

5. The solution of $\frac{2-x}{3x+1}=2$ is $x=0$ .

Solve for $x$ : $\frac{4}{3}x-6=7x+2$

1. We are given $\frac{4}{3}x-6=7x+2$ and are required to solve for $x$ .

2. We start with multiplying each of the terms in the equation by 3, then grouping like terms and then simplifying.

3. $\begin{array}{cccc}\hfill \frac{4}{3}x-6& =& 7x+2\hfill & \\ \hfill 4x-18& =& 21x+6\hfill & \left(\mathrm{each term is multiplied by}\phantom{\rule{3pt}{0ex}}3\right)\hfill \\ \hfill 4x-21x& =& 6+18\hfill & \left(\mathrm{rearrange}\right)\hfill \\ \hfill -17x& =& 24\hfill & \left(\mathrm{simplify grouped terms}\right)\hfill \\ \hfill \frac{-17}{-17}x& =& \frac{24}{-17}\hfill & \left(\mathrm{divide both sides by}\phantom{\rule{2pt}{0ex}}-17\right)\hfill \\ \hfill x& =& \frac{-24}{17}\hfill & \end{array}$
4. Substitute solution into original equation:

$\begin{array}{ccc}\hfill \frac{4}{3}Ã—\frac{-24}{17}-6& =& 7Ã—\frac{-24}{17}+2\hfill \\ \hfill \frac{4Ã—\left(-8\right)}{\left(17\right)}-6& =& \frac{7Ã—\left(-24\right)}{17}+2\hfill \\ \hfill \frac{\left(-32\right)}{17}-6& =& \frac{-168}{17}+2\hfill \\ \hfill \frac{-32-102}{17}& =& \frac{\left(-168\right)+34}{17}\hfill \\ \hfill \frac{-134}{17}& =& \frac{-134}{17}\hfill \end{array}$

Since both sides are equal to $\frac{-134}{17}$ , the answer is correct.

5. The solution of $\frac{4}{3}x-6=7x+2$ is,Â Â Â  $x=\frac{-24}{17}$ .

## Solving linear equations

1. Solve for $y$ : $2y-3=7$
2. Solve for $y$ : $-3y=0$
3. Solve for $y$ : $4y=16$
4. Solve for $y$ : $12y+0=144$
5. Solve for $y$ : $7+5y=62$
6. Solve for $x$ : $55=5x+\frac{3}{4}$
7. Solve for $x$ : $5x=3x+45$
8. Solve for $x$ : $23x-12=6+2x$
9. Solve for $x$ : $12-6x+34x=2x-24-64$
10. Solve for $x$ : $6x+3x=4-5\left(2x-3\right)$
11. Solve for $p$ : $18-2p=p+9$
12. Solve for $p$ : $\frac{4}{p}=\frac{16}{24}$
13. Solve for $p$ : $\frac{4}{1}=\frac{p}{2}$
14. Solve for $p$ : $-\left(-16-p\right)=13p-1$
15. Solve for $p$ : $6p-2+2p=-2+4p+8$
16. Solve for $f$ : $3f-10=10$
17. Solve for $f$ : $3f+16=4f-10$
18. Solve for $f$ : $10f+5+0=-2f+-3f+80$
19. Solve for $f$ : $8\left(f-4\right)=5\left(f-4\right)$
20. Solve for $f$ : $6=6\left(f+7\right)+5f$

if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2
wish i knew calculus to understand what's going on 🙂
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
24x^5
James
10x
Axmed
24X^5
Taieb
secA+tanA=2√5,sinA=?
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!