9th Class Mathematics Linear Equations in Two Variables Linear Equation in two Variables

Linear Equation in two Variables

Category : 9th Class

LINEAR EQUATION IN TWO VARIABLES

FUNDAMENTALS

An equation of the form \[ax+by+c=0,\]where a, b, c are real numbers \[(a\ne 0,b\ne 0)\] is called a linear equation in two variables x and y.

Example:- \[(i)\,\,3x+4y-2=0\]

\[\left( ii \right)\,\,4x+7y=3\]

Are the linear equations in x and y?

Solution:- Any point of values of x and y which satisfies the equation ax \[4x+by+=0\] is called a solution of it.

Example:- \[x=3,y=2\] is a solution of \[3x+2y=13\] because, when \[x=3\] and \[y=2,\] we have\[L.H.S=3\times 3+2\times 2=13=R.H.S\]

A linear equation in two variables has infinite solutions.
The graph of a two variables linear equation in two variables is a straight line.

Example:- \[x+2y=7\]

X	1	3	-1	-3
Y	3	2	4	5

The graph of equation parallel to y-axis The graph of equation parallel to x-axis

\[x=2\] \[y=2\]

Every point on the graph of a linear equation in two variables is a solution of the equation conversely every solution of the linear equation in two variable represents a point on the graph of the equation.
Let the two line represented by the equations \[{{x}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\,and\,{{x}_{2}}x+{{b}_{2}}y+c=0\]

Then, we have the following situations.

Number of solution	Condition	Nature of graph lines
(i) No solution	\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\]	Parallel
(ii) Unique solution	\[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\]	Intersect at a point
(iii) Infinitely many solution	\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\]	Coincident