Any linear relation of equality among variables is known as linear equation such as \(2x+1=3\), \(2x+3y=19\) or \(2x^2 + 3x+1=0\) etc.. Any equation of degree one is known as a linear equation that can be represented by \(ax + by + c = 0\), where \(a\) and \(b\) both are not zero simultaneously. For example if we say, cost of 3 pens and 4 books is Rs 100, in equation form this can be written as \(3p + 4b = 100\) where \(p\) is the price of one pen and \(b\) is the price of one book. This equation has 2 variables \(p\) and \(b\). Hence it is a linear equation with 2 variables. 

Equations can be used to solve various kind of puzzles and questions which are difficult to solve otherwise. Take this example: A man has three sons, first son has some coins, second son has 1 more than double the coins that the first son has, third son has 3 coins less than the second son, if the total number of coins with all the three sons is:

Let the first son has \(x\) coins, then number of coins with second and third sons will be: \(2x + 1\) and \((2x + 1) - 3\)

Sum of all the coins = \(x + 2x + 1 + (2x + 1 - 3) = 99\)

\( \Rightarrow 5x - 1 = 99\) or \(x = 20\)

Number of coins with the three sons = 20, 41 and 38  

One Variable Equation:

Linear equations involving only one variable are known as linear equations with one variable. 

For example \(3x + 4 = 19\) i.e. \(3x = 19 - 4 = 15\) 

or   \(3x = 15\)  or  \(x = 5\) 

Example  01: A student, instead of finding the value of seven eighth of a number, found the value of seven eighteenth of the number. If his answer differed from the actual one by 770, find the number.