Linear Equation: An equation in which the highest power of variable is one, i.e, degree of equation is one is called linear equation.

The general form of linear equation in two variables is ax+by+c=0; a0 and a, b, c are real numbers.

For e.g. : 2x+y=3; x+y=5; etc.

Note: In the linear equation ax+by+c=0; ‘x’ and ‘y’ are known as the variables.

The values of variables which satisfy the linear equations in two variables are called solutions of linear equations.

Note: A linear equation in two variables can have an infinite number of solutions.

Pair of Linear Equations in Two Variables

If we consider two linear equations at a time simultaneously then, such a pair is called a pair of linear equations.

e.g. 3x-5=0 and x+2y=3

Let us now do a simple question on how to form a linear equation in two variables from a given problem statement?

Q. Raj tells his daughter, “seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Represent this situation in the form of a linear equation.

Sol. Let present age of Raj is x years

       and present age of his daughter be y years

Case I. 7 years ago,

            Age of Raj and his daughter will be (x-7) and (y-7) years respectively.

            According to question,

            (x-7) = 7(y-7)

            x-7 = 7y - 49

            x-7y = -42

Case II. 3 years from now,

             Age of Raj and his daughter will be (x+3) and (y+3) years respectively.

             According to question,

              (x+3) = 3(y+3)

              x+3 = 3y+9

              x-3y = 6 

So, x-7y = -42 and  x-3y = 6 are the required pair of linear equations in two variables.

Consistency of linear equation

• The equations which have solutions are called consistent equations and those equations which do not have solutions are called inconsistent equations.

• If the equation has a solution, then it has either unique solution or infinite solutions.

Conditions for Consistency of linear equations

If a1x+b1y+c1 = 0

   a2x+b2y+c2 = 0

   are the given linear equations in two variables, then

Q. On comparing the ratios, find out whether the following pair of linear equations are consistent or inconsistent.

i) 3x+2y = 5; 2x-3y = 7

Sol. Compare the given equation with a1x+b1y = c1 and  a2x+b2y = c2

    ∴ a1a2 = 32  and  b1b2  = 2-3

       Clearly, a1a2 b1b2

       Given equation represents a unique solution, that is, they are consistent.

ii) x+y = 5; 2x+2y =10

Sol. Compare the given equation with a1x+b1y = c1 and  a2x+b2y = c2

     ∴a1a2 = 12;  b1b2  = 12;  and   c1c2 = 12

        Clearly,  a1a2 = b1b2 = c1c2

        It has infinite solutions i.e. they are consistent.

Now, we have learnt almost everything about what a linear equation is. It’s time to dive into the methods on how to solve linear equations using different methods.

Algebraic Methods of Solving a Pair of Linear Solution

I. Substitution Method 

The substitution method is a simple way to solve a system of linear equations algebraically and find the solutions of the variables. As the name suggests, it involves finding the value of x-variable in terms of y-variable from the first equation and then substituting the value of x-variable in the second equation. In this way, we can solve and find the value of the y-variable. And at last, we use this value of y in any of the given equations to find x.

To solve a linear equation using the substitution method, the following steps are taken.

• Simplify the given equation if needed.

• Solve any one of the equations for any one of the variables. You can choose any variable.

• Substitute the obtained value of x or y in the other equation.

• Simplify the new equation obtained using arithmetic and solve the equation for one variable.

• Now, substitute the value of the variable from above step in any of the given equations to solve for the other variable.

Let us take an example of solving two equations x+y=14 and x-y=4 using the substitution method.

Given equations are:

                                 x+y=14       _______(1)

                                 x-y=4          _______(2)

                    from equation (1),

                                 x = 14-y      _______(3)

                    On putting value of x in equation (2), we get

                             14-y-y = 4

                             14-2y = 4

                              -2y = -10

                              y = 5

                     from equation (3)

                             x = 14 -y

                             x = 14 -5

                             x = 9

                 ∴ Solution of given equation is y=5 and x=9

Another question on substitution method

II. Elimination Method

The elimination method of solving a system of linear equations algebraically is the most widely used method out of all the methods to solve linear equations. In the elimination method, we eliminate any one of the variables by using basic arithmetic operations and then simplify the equation to find the value of the other variable.

To solve a linear equation using the elimination method, the following steps are taken.

• The first step is to multiply or divide the linear equations with a non-zero number to get a common coefficient of any of the variables in both equations.

• Add or subtract both equations so that the same terms will get eliminated.

• Simplify the result to get a final answer of the left-out variable(say y) such that we will only get a solution of the form y=c, where c is any constant.

• At last, substitute this value in any of the given equations to find the value of the other variable.

Let us take an example of two linear equations x+y=5 and 2x-3y=4 to understand it better.

Given equations are:

                                x+y = 5        _______(1)

                                2x-3y = 4     _______(2)

Multiplying Eq.(1) by 3 and Eq.(2) by 1

                                3x+3y = 15  _______(3)

                                2x-3y = 4     _______(4)

Now, add Eq.(3) and Eq.(4)

                                3x+3y = 15

                               + 2x-3y = 4   

                           ------------------------

                                   5x = 19

                                x = 195

By putting the value of x in Eq.(1)

                                  x+y = 5

                              195+y = 5

                            y = 5 - 195

                            y = 65

  Hence, values are x=195 and y=65

One more question on elimination method(problem statement)

III. Cross multiplication Method

Cross multiplication method is used in solving linear equations in two variables. The simplest and easiest method of solving linear equations in two variables is done by the method of cross-multiplication. 

Cross Multiplication Formula:

If a1x+b1y+c1 = 0  and  a2x+b2y+c2 = 0 are the given pair of linear equations then,

using the cross multiplication formula, the values of x and y is given by:

As we already have a formula, let us see how to solve linear equations with two variables by taking an example.

Q. Suppose that we have to solve the following pair of equations:

                                       8x+5y = 9   ________(1)

                                       3x+2y = 4   ________(2)

On comparing the given equations with a1x+b1y+c1 = 0 and   a2x+b2y+c2 = 0