In this section we will discuss Linear Equation in two variables and different ways of solving it.
Linear equation of the form ax + by + c = 0 or ax + by = c, where a, b and c are coefficient of x and y respectively and they are real numbers, 'a' is not equal to zero (a≠0) ,'b' is not equal to zero(b ≠ 0) and x and y are variables , is called a Linear equations in two variables.
Example 1 : 2x + 3y - 4 = 0
Here, a = 2 ; b = 3 and c = -4. Example 2 : -x + y = 0
Here, a = -1 ; b = 1 and c = 0
A pair of linear equation in two variables is said to form a system of simultaneous linear equation.
The values of x and y are the solution of the given equation. These values satisfies the given equation. Consistent : If there is at least one solution then the equations are consistent. In –consistent : If there is no solution then the equations are in-consistent.
If the two lines are a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 then
2)Coincident lines, if (infinitely many solutions)
3)Parallel lines, if = In-consistent (No solution)
Example 1: For what value of k will the equations x + 2y + 7 = 0;
2x + ky + 14 = 0 represent the coincident line. Solution : For coincident lines we have,
⇒ 1 / 2 = 2 / k = 7 / 14
⇒ 1 /2 = 2 / k
∴ k = 4
There are 4 ways to solve the linear equations.