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Pair of Linear Equations in Two Variables


Linear Equation in Two Variables
A linear equation in two variables is an equation of the form ax + by + c = 0 where a ¹ 0, b ¹ 0 and c are constants ( real numbers) and x, y are variables.
 
Example

x + 2y = 3, 2x - 3y - 5 = 0

Solution

A pair of values of x and y satisfying the equation ax + by + c = 0 is known as the solution of the equation.

 

Example : x = 1, y = 2 is a solution of x + 3y = 7

 

Graph
When we draw the graph of the equation
ax + by +c = 0 we get a straight line. 

Note
The graph of a first degree equation in one or two variables is a (straight) line. Hence we call it a linear equation.
 

Example

Draw the graph of y = 2x - 1. Use it to find the solutions of the equation. Check from the graph whether x = -1, y = -3 is a solution.

Solution

y = 2x - 1

If x = 0; y = (2 × 0) -1 = -1
If x = 1; y = (2×1) -1 =1
If x = 2; y = (2×2) -1 =3
We thus have the following points. 

x   0   1  2
y  -1   1  3

Plotting A (0, -1), B (1,1), C (2,3), we draw the graph.


 


We see that the point P (-1, -3) lies on the line, which shows that x = -1 and y = -3 is a solution of the equation y = 2x - 1. Also Q (-3/2, -4) and R (-2,-5) lie on the graph indicating that x = -3/2, y = -4 and x = -2,y = -5 are the other solutions of the above equation.
 




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