Introduction:
What is an equation?.
An equation is a statement in which two algebraic expressions are equal. Linear Equations of one variable is the equation which involves only one variable. General form of a linear equation of one variable is ax + b = 0, where a is not equal to 0.
Let us discuss the situation that " A farmer traveled a distance of 61 km in 9 hrs. He traveled partly on foot at an average of 4 km / hr and partly on bicycle at an average speed of 9 km/hr. Find the distance traveled by him on foot ". If we look at this situation it seems to be complicated. But we can easily solve this types of situations framing linear equations of one variable.
Solving linear equations in one variable
Linear Equation: The general form of Linear Equations are ax + b = 0, where a is not equal to 0.
Solution of a Linear Equation: The value of the variable which when substituted for the variable in the equation, makes its two sides equal, is called a solution ( 0r root) of the equation.
Example 1 : Verify that x = 2 is a solution of the linear equation 2x + 7 = 13  x
Solution: Substituting x = 2 in 2x + 7 = 13  x, we get
2 ( 2) + 7 = 13  2
=> 4 + 7 = 11
=> 11 = 11
=>
x=2 is the solution of the given equation.Example 2: Verify that x = 3 is not a solution of the equation, 3x  5 = 4 + x
Solution: Substituting x = 3, in 3x  5 = 2 + x, we get,
3 (3)  5 = 2 + 3
=> 9  5 = 5
=> 4 = 5, which is not true
Therefore,
x = 3 is not a solution of the given equation.
How to solve Linear Equations in one variable: Rules for solving linear equations:The equality of a linear equation is not changed,
Rule 1 : When the same number is added to both sides of the equation.
Rule 2: When the same number is subtracted from both sides of the equation.
Rule 3 : When both sides of the equation are multiplied by the same nonzero numbers.
Rule 4: When both sides of the equation are divided by the same nonzero number.
What is Transposition? Any term of an equation may be taken to the other side with the sign changed, without affecting the inequality.
This process is called transposition.
Examples of linear equation in one variable:
Example : Solve : 5x + 4 = 4x  10
Solution: 5x + 4 = 4x  10
=> 5x  4x =  10  4
=> x =  14
Cross Multiplication:
If $\frac{a}{b}= \frac{c}{d}$, then a x d = b x c
This process is called Cross Multiplication.
Linear equations in one variable word problems:
Method of solving liner equations in one variable word problems we should follow the following steps.
Step 1: Read the problem carefully and find out what is given and what is unknown.
Step 2: Represent the unknown quantity by x.
Step 3: Frame an equation in x, as per the conditions given in the problem.
Step 4 : Solve for x.
Example : A chemist has one solution containing 50% acid and a second one containing 25% acid. How much of each should be mixed to make 10 litres of a 40$ acid solution?
Solution: It is given that the total number of litres of the mixture = 10 lit
Let us fill up the following table.

Solution 1

Solution 2 
Concentration 
50% 
25% 
No. of litres 
x

( 10  x)

Quantity of acid in 50% of the solution = 50% x = 0.5 x
Quantity of acid in 25% of the solution = 25% of ( 10  x) = 0.25 ( 10  x)
Quantity of acid in 40% of the mixture = 40 % of 10 = 0.40 x 10 = 4
Hence we have the equation,
0.5 x + 0.25 ( 10  x ) = 4
=> 0.5 x + 0.25 x 10  0.25 x = 4
=> 0. 5 x  0.25 x + 2.5 = 4
=> 0.25 x + 2.5  2.5 = 4  2.5
=> 0.25 x = 1.5
=> x = 1.5/0/25
= 6 litres
Therefore the scientist should mix 6 litres of 50% solution and 4 litres of 25 % solution.
How to solve linear equation in one varaible:
Linear equations in one variable can be written by taking an example of 2x+3=6.
Solution
2x=63=3
2x=3
x=3/2.
Linear equations in one variable examples
Solve ( 5x  7) = 8
Solution: We have 5x  7 = 8
=> 5x = 8 + 7 [ when we transpose 7 to the other side it becomes + 7 ]
= 15
=> x = 15 / 5 = 3
Hence the solution
Example 2: Solve : $\frac{6x  7}{3x + 1}\: = \: \frac{2x+1}{x+5}$
Solution: We have $\frac{6x  7}{3x + 1}\: = \: \frac{2x+1}{x+5}$
=> ( 6x  7 ) ( x + 5) = ( 2x + 1) ( 3x + 1) [ by cross multiplication ]
=> 6x
^{2} +23 x  35 = 6 x
^{2} + 5x + 1 [ Using FOIL ]
=> 6 x
^{2}  6 x
^{2} + 23 x  5 x  35 = 6x
^{2}  6 x
^{2} + 5x + 1 5 x [adding 6x
^{2}  5x on both sides ]
=> 8 x  35 = 1
=> 18 x  35 + 35 = 1 + 35 [ adding 35 on both sides ]
=> 18 x = 36
[ dividing both sides by 18 ]
=> x = 36/18 = 2
The solution to the above equation is x = 2