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Find the value of the variable in given linear equations :
i) 7x – 5 = 2x
ii) 5x – 12 = 2x – 6
iii) 7p – 3 = 3p + 8

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Hint: Consider all the linear equations separately, make the variables as subjects or take variables to one side and the constant terms on the other side and hence find the value of the variable.

Complete step-by-step answer:

We are given with linear equation to find the values of variables which are:
i) 7x – 5 = 2x
ii) 5x – 12 = 2x – 6
iii) 7p – 3 = 3p + 8
First we will understand what linear equations are and learn some facts about them.
Let a linear equation be written in the form of ax + b = 0.
Where ‘a’ and ‘b’ are real numbers and ‘x’ is a variable. This is sometimes called the standard form of linear equations. Please note that most linear equations will not start of this form. Also the variable may or may not be as ‘x’ so please don’t get too locked into always seeing ‘x’ there.

Now for solving linear equations we will make heavy use of facts which are,
i) If a = b then a+c = b+c for any value c. This means that we can add a number ‘c’ to both the sides of the equation and the value of the equation does not change.
ii) If a = b then a-c = b-c for any value c. This means that we can subtract a number ‘c’ from both the sides of the equation and the value of the equation does not change.
iii) If a = b then ac = bc for any non-zero value of c, so that the value of the equation remains unaltered.
iv) If a = b then $\dfrac{a}{b}=\dfrac{b}{c}$ for any non-zero value of c, so that the value of the equation remains unaltered.
These points are very important and help very much while solving any liner type of equation. They should be kept in mind.

i) So the 1st linear equation given as,
7x – 5 = 2x………….(i)
Now subtracting ‘2x’ from both the side of equation (i) we get,
5x – 5 = 0……………..(ii)
Now adding ‘5’ to both the sides of equation (ii) we get,
5x = 5
So the value of x is 1.

ii) The 2nd linear equation given is
5x – 12 = 2x – 6……………..(iii)
Now adding 12 to both sides of equation (iii) we get,
5x = 2x + 6…………………….(iv)
Now subtracting 2x from both the sides we get,
3x = 6
Dividing throughout by ‘3’, we get the value of x as 2.

iii) So, the 3rd linear equation given is
7p – 3 = 3p + 8…………….(v)
Now adding 3 to both sides of equation (v) we get,
7p = 3p + 11……………..(vi)
Now subtracting 3p from both the sides of equation (vi) we get,
4p = 11
So the value of p is $\dfrac{11}{4}$ .

Note: Students confuse themselves if they see any other variable other than x as they don’t have a habit to see it. Also they should follow all the rules of linear equations. Another simpler approach is just bringing all the variables to the left side and known values to the right side and solving accordingly.
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