How to solve linear equations with fractions?

 

How to solve linear equations with fractions:-

Today I am trying to explain a nice topic,
How to solve linear equations with fractions.
Let's first know before i start my topic.

**What is a linear equation?

Answer:-

An equation which has one or two variables and highest power of the variables are one and when plotted on the graph forms a straight line is called a linear equation.
ax + y = 0 is a linear equation of two variables, where x, y are two variables.
again,
ax + 10 = 0 is a linear equation of one / single variable of x.

The standard form of a linear equation is

We'll learn now,

**What is  linear equation with fractions?

Answer:-

A linear equation which has a fractional number / fraction together with its variable is known as a linear equation with fractions.
e.g; (i) ⅔ x + 7 =0
(ii) ⅚ y - 12 = ½ y
(iii) ⅕ x + ¾ y = 9
……… and many more.
Now I will start our main motive discussion about how to solve linear equations with fractions.
To start the discussion,

Let's take a linear equation of one variable and we'll learn, How do you solve a linear equation with fractions on both sides?

Solve :-

**Question(i):-

Solution:- 

1st method:-

⅖ x + ⅚ = ⅓ x + ⅗ 
= ⅖ x - ⅓ x = ⅗ - ⅚ 
=(3 . 2 - 5 .1) x / 15 = (6×3 - 5×5) /30 [ ∵ L.C.M of 5,3 is 15 and that of 5,6 is 30]
= (6 - 5)x /15 = (18 - 25) / 30
= x / 15 = - 7 / 30
= x = - 7 × 15 / 30
= x = - 7/ 2 solved.

2nd method :-

⅖ x + ⅚ = ⅓ x + ⅗ 
=;⅖ x - ⅓ x = ⅗ - ⅚ 
= 30(⅖ x - ⅓ x) = 30 (⅗ - ⅚) [ L.C.M of 5,6,3 and 5 is 30] 
=>30 . 2 . x / 5 - 30.1x / 3 = 30.3 /5 - 30. ⅚
=> 6.2 x - 10x = 6 .3 - 5.5
=> 12x - 10x = 18 - 25
=> 2x = - 7
=> x = -7/2 Solved.

Rule (1):- 

For the first method of question (i):
(i) 1st the fractions with variable will be taken in either side (here we have taken them in the left hand side) and changed their signs.
(ii) L.C.M of the denominators of the fractions of both sides are taken
(iii) L.C.Ms are divided by the denominators and the result of the division is multiplied by the numerator in both sides.
(iv) the numbers in the numerator are subtracted on both sides.
(v) if the denominator of the lLeft hand side is multiplied by the fraction of the right hand side, then the final answer comes.

Rule (2):-

For the second method of question (i):

(i) 1st the fractions with variable variable will be taken in either side (here we have taken them in the left hand side) and will change their signs.
(ii) L.C.M of all the denominators of both sides is taken (here the denominators of both sides are 5,3,5 and 6) and their L.C.M is 30
(iii) Now multiply 30 individually by all the fractions on both sides.
(iv) when the numbers are multiplied, and the result is subtracted from bigger to smaller number, then the final result is reached.

More Questions - Solving an equation in One Variable with Fractions

** Question (ii):- ⅜ y - 5 = ¼ y

Solution :- 

1st method :-

⅜ y - 5 = ¼ y
=>⅜ y - ¼ y = 5
=>(3×1×y - 2 ×1 × y)/8 = 5 [∵ L.C.M of 4 and 8 is 8]
=>(3y - 2y)/8 = 5
=> y = 5 × 8
=> y = 40 solved.

2nd method:-

⅜ y - 5 = ¼ y
=>!⅜ y - ¼ y = 5
=> 8 (⅜ y - ¼ y) = 8 × 5
=> (8 × 3 × y)/ 8 - (8 × 1 × y)/ 4 = 40
=> 3y - 2y = 40
=> y = 40 Solved

Rule (3):- 

For 1st Method of Question (ii):-

(i) 1st the fractions with variable will be taken in either side (here we have taken them in the left hand side) and changed their signs.

(ii) L.C.M of the denominators of L.H.S fractions is taken.
L.C.M is divided by the denominators and the result of the division is multiplied by the numerator in the left hand side.
(iv) the numbers in the numerator are subtracted on both sides.
(v) the denominator of the left hand side is multiplied by the number (5) of the right hand sides, then the final answer comes.

Rule (4) :-

2nd method of Question (ii):-
(i) 1st the fractions with variable will be taken in either side (here we have taken them in the left hand side) and changed their signs.
(ii) L.C.M of the fractions of the left hand side
8 is 8, so 8 is multiplied on both sides and each individual term of the sides is multiplied.
(iii) then, for each term with variable y, the final result is the sum of the two.

***question (iii):- ⅝ x + y = 12

Solution: - This is a single linear equation of two varies, so it has infinitely many solutions to solve, so we can solve this using the following method.

⅝ x + y = 12……. . (i)
Let's go,
x = 8
Now (i) => (5 × 8)/ 8 + y = 12 [ ∵ putting the value of x]
=> 5 + y = 12
=> y = 12 -5
=> y = 7
Let's go,
x = - 8
Now from (i) => 5 × (-8)/8 + y = 12
=> - 5 + y = 12
=> y = 12 + 5
=>y = 17

………………………………………………
……………………………………………….
And many more solution we can find taking different values of x gives different values of y.

Rule (5):-

(i) we have assumed a value of x here we have taken 8 and have put this value in the place of x in the equation now we have 8 (in the numerator) with 8 of the denominator.
(ii) Now 5 is taken in R.HS after changing its sign, then 5 is subtracted from 12 to get y = 7
       ∴ Required solution of the equation is,
               [ x = 8        x = - 8    ]
               [ y = 7        y =   17  ]  Solved

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FREQUENTLY ASKED QUESTIONS:-

Write the standard form of a linear equation.

Answer:- The standard form of linear equation is :- ax + by + c = 0, where x and y are two variables and a,b and c are literal fectors.

Write the standard form of a quadratic equation.

Answer :-The standard form of a quadratic equation is :- ax² + bx + c = 0, where a,b and c are literal fectors.Again a,b are called the co-efficients of x² and x terms respectively.

What is a linear equation ?

Answer:- An equation which has one or two variables and highest power of the varibles are one and when plotted on the graph forms a straight line is called a linear equation. ax + y = 0 is a linear equation of two variables, where x, y are two variables. again, ax + 10 = 0 is a linear equation of one / single variable of x.

what is the degree of a linear equation?

Answer :- The degree of a linear equation is one (1).

How many solutions can have a lone linear equation of two variable ?

Answer:- Infinitely.

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