Linear Equations in Two Variables

An equation of the form ax + by + c = 0, where a,b,c ϵ R , a≠0, b≠0 and x,y are variables is called a linear variable s or an equation of first degree in two variables.

Some examples of linear equations in two variables are:

3x + 7y =9                   ,                                  2x – y = 0 

5/2 u - 9/13 v + 2 =0 , √3 t + (√5)/2 z = 7

Where x ,y; t and z are variables.

SOLUTION OF LINEAR EQUATION IN TWO VARIABLES

A pair of value of x and y which satisfies the equation ax + by + c = 0, where a, b, c ϵ R , a≠0, b≠0 is called the solution of ax + by + c = 0.

Example: the equation 2x + y + 4

Solution: 

Express y in terms of x

Put any value of x and find the corresponding value of y

The values of x and y so obtained give solution of the equation 

Here 2x + y = 4

Y = 4 – 2x

Put x =0, then y = 0; x=0 and y = 4 or the ordered pair (0,4) is the solution of the given equation.

Next, put x=1, then y = 4 –(2×1) =4 – 2 =2; thus x = 1 and y = 2 or the ordered pair (1,2) is another solution of the given equation.

Let’s observe some example:

Solve: 7x + 2y =23

           X – Y=2

Solution: the given equations are

7x +2y = 23 …………….. (i)

X –y = 2 …………….. (ii)

From (ii), we get y = x – 2 ………………… (iii)

Putting (iii) in (i), we have

7x + 2(x-2) = 23

Or 7x + 2x – 4 = 23

Or 9x = 23 + 4 =27

Therefore, x = 27/9 = 3 ………………. (iv)

Putting (iv) in (ii), we have

3 –y = 2

3 – 2 = y

Therefore, x =3, y = 1 is the required solution.

Solve the system of homogeneous linear equations.

                      Q1.   2x – 5y = 0 and 19x + y = 0

Q2. Solve 2x-3y =13 and 7x-2y =20

Q3. Solve :

                     5x-24y =16

                     4x-y     =31

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