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Linear Equations in Two Variables - Class 9 : Notes


 

(1) An equation of the form ax + by + c = 0, where a, b, c are real numbers such that a and b not both zero is called a linear equation in two variables.
For Example: 2x + 3y + 5 = 0, where a = 2, b = 3 which are not zero. So this is linear equation in two variables.

(2) A linear equation in two variables has infinitely many solutions.
For Example: we have 4x - y - 3 = 0, there is many solution
If we take x = 1 , y = 1 then LHS = 4 \times 1 - 1 - 3 = 0 = RHS
If we take x = 2 , y = 5 then LHS = 4 \times 2 - 5 - 3 = 0 = RHS

(3) The graph of a linear equation in two variables is a straight line.
For Example:
x + y = 4
We have x + y = 4 \Rightarrow y = 4 - x
When x = 0, we have: y = 4 - 0 = 4
When x = 2, we have: y = 4 - 2 = 2
When x = 4, we have: y = 4 - 4 = 0
Thus, we have the following table:Plotting the points (0, 4) (2, 2) and (4, 0) on the graph paper and drawing a line joining them.

(4) The equations of x and y-axes are y = 0 and x = 0 respectively.
For Example:
x + 3 = 0 equation is for x-axes because in this equation y = 0.
7y - 3 = 0 equation is for y-axes because in this equation x = 0.

(5) The graph of the equation x = a is a straight line parallel to y-axis.
For Example:
The equation for such a line x = - {9 \over 2} is given below:

(6) The graph of the equation y = a is a straight line parallel to x-axis.
For Example: For a line that is parallel to the x-axis, the equation for such a line y = 2 is given below:

(7) Every point on the graph of a linear equation in two variables is a solution of the equation. Conversely, every solution of linear equation in two variables represents a point on the graph of the equation.
For Example: In linear equation f(x) = x + 2y - 1 = 0 , \left( {3, - 1} \right) also lies on this line.



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