Linear Equation in two variables
An equation of the form ax + by + c = 0 or ax + by =c, where a,b, c are real numbers, a ≠ 0, b ≠ 0 and x, y are variables, is called a linear equation in two variables.
Examples of linear equation in two variables :
• x + 2y = 1
• -2x + 3y = 4
Solution of Linear equation in two variables
Let ax + by + c = 0, where a, b, c are real numbers, a ≠ 0, b ≠ 0. Then, any pair of values of x and y which satisfies the equation ax + by + c = 0, is called a solution of it.
Example: x = 3, y = 2 is a solution of 3x – 2y = 5 because when x = 3,
y = 2, we have: LHS = 3 x 3 – 2 x 2 = 5 = RHS.
But, x = 3, y = -2 is not its solution, because 3 x 3 – 2 x (-2) ≠ 5 i.e.
LHS ≠ RHS when x = 3 and y = -2.
Graph of a Linear Equation in two variables
In order to draw the graph of a linear equation ax + by + c = 0, a ≠ 0,
b ≠ 0, we follow the steps written below :
Step I – Obtain the linear equation, Let the equation be ax + by + c = 0.
Step II – Express y in terms of x to obtain y =
Step III – Give any two values to x and calculate the corresponding values of y from the expression in step II to obtain two solutions, say (α1,β1) and (α2, β2).
If possible take values of x as integers in such a manner that the corresponding values of y are also integers.
Step IV – Plot points (α1,β1) and (α2, β2) on a graph paper.
Step V – Join the points marked in step IV to obtain a line. The line obtained is the graph of the equation ax + by + c = 0.
Important Remarks
(i) When a ≠ 0, c ≠ 0 and b = 0
The equation ax +by + c = 0, reduces to ax + c = 0 or x = -c/a. In this case the graph of the equation ax + by + c = 0, is a straight line parallel to
y – axis and passing through the point (-c/a, 0)
(ii) When b ≠ 0, c ≠ 0 and a = 0.
In this case, the equation ax + by + c = 0, is a straight line parallel to
x – axis and passing through the point (0, -c/b).
(iii) When a ≠ 0, b = 0, c = 0.
The equation ax + by + c = 0 reduces to ax = 0 i.e. x = 0. The graph of this equation is y – axis.
(iv) When a = 0, b ≠ 0, c = 0.
The equation ax + by + c = 0 reduces to by = 0 i.e. y = 0. The graph of this equation is x – axis.
(v) When c = 0.
The equation ax + by + c= 0 reduces to ax + by = 0. The graph of this equation is a line passing through the origin.
To get a more clear idea, let’s explain with an example :
Example : Draw the graph of the equation y – x = 2.
We have, y – x = 2
y = x + 2
When x = 1, we have: y = 1 + 2 = 3
When x = 3, we have: y = 3 + 2 = 5
Thus, we have the following table exhibiting the abscissae and ordinates of points on the line represented by the given equation.
Plotting the points (1, 3) and (3, 5) on the graph paper and drawing a line joining them, we obtain the graph of the line represented by the given equation as shown :
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