Revision Notes for CBSE Class 9 Maths Chapter 4 - Linear Equations in Two Variables

• Any equation which can be written in the form $ax+by+c=0$ , where $a,b$ and $c$ are real numbers $a\ne 0$, $b\ne 0$ is called a linear equation in two variables.

• An ordered pair $\left( x,y \right)$ is the solution of linear equation in two variable if this point satisfies the linear equation $ax+by+c=0$.

• Examples of linear equation in two variables - $2x+4y=1,x-10y=-5$, etc.

Solution of Linear Equation:

• A linear equation has a unique solution when there exist only one point which satisfies the linear equation.

For example: Solution of $2x+6=2$ is

$2x+6=2$

$2x=2-6$ 

$2x=-4$ 

$x=\dfrac{-4}{2}$ 

$x=-2$ 

In $2x+6=2$ has only one variable $x$ therefore $x$ has unique solution. Also, geometrically it will be a point on rectangular axes whose ordinate will be $0$ 

• A system of linear equation has unique solution when the system of lines intersects each other at only one point.

• A linear equation in two variables have infinitely many solutions means there are more than one ordered pair which satisfy the equation.

• For example: Solution of $2x+3y=12$ are

The following value $\left( 3,2 \right),\left( 0,4 \right),\left( 6,0 \right)$ of $x$ and $y$ satisfies the equation $2x+3y=12$ therefore they are the solutions of $2x+3y=12$.

• A system of linear equation has infinitely many solution if the system of lines coincides each other which means each point on the system of line will be the solution.

• For example: System of linear equations $-6x+4y=2$ and $3x-2y=-1$ have infinitely many solution because these two lines coincide each other as shown in graph below 

Graph of Linear Equation in Two Variables: 

• We know that linear equation in two variables can have infinitely many solutions and we get every solution in form of pair of values.

• So, we can plot these values on coordinate plane and draw the graph of linear equation in two variables. 

     For e.g. – Let us draw the graph for the equation $x+y=2$ 

    Let us draw a table for the values of $x$  and $y$

Now, Plotting the values of $x$  and $y$ in the coordinate plane  

• From the above graph we can see that geometrical representation of given equation is a straight line.

Equations of Line Parallel to X-axis and Y-axis: 

• Linear equation in two variables is written as $ax+by+c=0$ if we put $y=0$, the equation becomes $ax+c=0$. The Graph of equation $ax+c=0$ is a straight line parallel to the y-axis.

• On the other hand, if we put $x=0$ in $ax+by+c=0$, the equation becomes $by+c=0$.The Graph of equation $by+c=0$ is a straight line parallel to the x-axis.

• Equation of x-axis is $y=0$ because at x-axis y-coordinates are always zero and the coordinate form of any point on x-axis will be $\left( x,0 \right)$  

• Equation of y-axis is $x=0$ because at y-axis x-coordinates are always zero and the coordinate form of any point on y-axis will be $\left( 0,y \right)$  

• Graph below represents the equation of x-axis and y-axis

• If in a coordinate point $\left( x,y \right)$ value of $x$ is a positive constant then the point will lie on the right side of y-axis and if it is a negative constant then the point will lie on the left side of y-axis.

• Similarly, if the value of $y$ is a positive constant then the point will lie on the upper side of x-axis and if it is negative constant then the point will lie on the lower side of x-axis.