How to Solve Linear Equations?

How to solve linear equations?

Step-by-step instructions are given in the examples of solving linear equations. We will learn how to solve one variable linear equations using addition, subtraction, multiplication and division.


Examples on solving linear equations: 

1. Solve the equation 2x - 1 = 14 - x and represent the solution graphically. 

Solution: 

2x - 1 = 14 - x 

⇒ 2x + x = 14 + 1

(Transfer -x from right hand side to the left hand side, then negative x changes to positive x. Similarly again transfer -1 from left hand side to the right hand side, then negative 1 change to positive 1.


Therefore, we arranged the variables in one side and the numbers in the other side.)

⇒ 3x = 15

⇒ 3x/3 = 15/3 (Divide both sides by 3)

⇒ x = 5

Therefore, x = 5 is the solution of the given equation.

The solution may be represented graphically on the number line by graphing linear equations.

graphing linear equations



2. Solve the equation 10x = 5x + 1/2 and represent the solution graphically.

Solution:

10x = 5x + 1/2

⇒ 10x – 5x = 1/2

(Transfer 5x from right hand side to the left hand side, then positive 5x changes to negative 5x).

⇒ 5x = 1/2

⇒ 5x/5 = 1/2 ÷ 5 (Divide both sides by 5)

⇒ x = 1/2 × 1/5

⇒ x = 1/10

Therefore, x = 1/10 is the solution of the given equation.


The solution may be represented graphically on the number line.

solution graphically



3. Solve the equation 6(3x + 2) + 5(7x - 6) - 12x = 5(6x - 1) + 6(x - 3) and verify your answer

Solution:

6(3x + 2) + 5(7x - 6) - 12x = 5(6x - 1) + 6(x - 3)

⇒ 18x + 12 + 35x - 30 - 12x = 30x - 5 + 6x - 18

⇒ 18x + 35x - 12x + 12 - 30 = 30x + 6x - 5 - 18

⇒ 41x - 18 = 36x - 23

⇒ 41x - 36x = - 23 + 18

⇒ 5x = -5

⇒ x = -5/5

⇒ x = -1

Therefore, x = -1 is the solution of the given equation.

Now we will verify both the sides of the equation,

6(3x + 2) + 5(7x - 6) - 12x = 5(6x - 1) + 6(x - 3) are equal to each other;

Verification:

L.H.S. = 6(3x + 2) + 5(7x - 6) - 12x

Plug the value of x = -1 we get;

= 6[3 × (-1) + 2] + 5 [7 × (-1) - 6] - 12 × (-1)

= 6[-3 + 2] + 5[-7 - 6] + 12

= 6 × (-1) + 5 (-13) + 12

= - 6 - 65 + 12

= -71 + 12

= -59

Verification:

R.H.S. = 5(6x - 1) + 6(x - 3)

Plug the value of x = - 1, we get

= 5[6 × (-1) - 1] + 6[(-1) - 3]

= 5(-6 - 1) + 6(-1 -3)

= 5 × (-7) + 6 × (-4)

= - 35 - 24

= - 59

Since, L.H.S. = R.H.S. hence verified.

What is cross multiplication?

The process of multiplying the numerator on the left hand side with the denominator on the right hand side and multiplying the denominator on left hand side with the numerator on right hand side is called cross multiplication.

And then equating both the products we get the linear equation.

On solving it we get the value of variable for which L.H.S. = R.H.S. Then, it is an equation of the form.

(mx + n)/(ox + p) = q/r where m, n, o, p, q, r are numbers and ox + p ≠ 0
⇒ r(mx + n) = q(ox + p)

It’s an equation in one variable x but it is not a linear equation as L.H.S. is not a linear polynomial.

We convert this into linear equation by the method of cross multiplication and further solve it step-by-step.


Examples on cross multiplication while solving linear equations:

1. (3x + 4)/5 = (2x - 3)/3

Solution:

(3x + 4)/5 = (2x - 3)/3

On cross multiplication, we get;

⇒ 3(3x + 4) = 5(2x - 3)

⇒ 9x + 12 = 10x - 15

⇒ 9x - 10x = -15 - 12

⇒ -x = -27

⇒ x = 27

Verification:

L.H.S. = (3x + 4)/5

Plug x = 27, we get;

(3 × 27 + 4)/5

= 81 + 4/5

= 85/5

= 17

Verification:

R.H.S. = (2x - 3)/3

Plug x = 27, we get;

(2 × 27 - 3)/3

= 54 - 3/3

= 51/3

= 17

Since, L.H.S. = R.H.S. hence verified.


2. Solve 0.8 - 0.28x = 1.16 - 0.6x

Solution:

0.8 - 0.28x = 1.16 - 0.6x

⇒ 0.6x - 0.28x = 1.16 - 0.8

⇒ 0.32x = 0.36

⇒ x = 0.36/0.32

⇒ x = 36/32

⇒ x = 9/8

Therefore, 9/8 is the required solution.

Verification:

L.H.S. = 0.8 - 0.28x

Plug x = 9/8, we get;

= 0.8 - 0.28 × 9/8

= 8/10 - 2̶8̶/100 × 9/8̶

= 8/10 - 63/200

= (160 - 63)/200

= 97/200

Verification:

R.H.S. = 1.16 - 0.6x

= 1.16 - 0.6 × 9/8

= 116/100 - 6̶/10 × 9/8̶

= 116/100 - 27/40

= (232 - 135)/200

= 97/200

Since, L.H.S. = R.H.S. hence verified.


 Equations

What is an Equation?

What is a Linear Equation?

How to Solve Linear Equations?

Solving Linear Equations

Problems on Linear Equations in One Variable

Word Problems on Linear Equations in One Variable

Practice Test on Linear Equations

Practice Test on Word Problems on Linear Equations


 Equations - Worksheets

Worksheet on Linear Equations

Worksheet on Word Problems on Linear Equation









7th Grade Math Problems

8th Grade Math Practice 

From How to Solve Linear Equations? to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More