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.

Equation (1) a1= b1= c1=


Equation (2) a2= b2= c2=

X= Y=


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
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

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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