Linear Equation in two variables

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 a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 then

1) Intersecting lines,if = consistent (one solution) 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

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There are 4 ways to solve the linear equations.

Linear equation in two variables

• Solving linear equation by graphical method.
• Substitution method.
• Solving system of equation by elimination method
• Cross multiplication method or Cramer’s rule

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