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

Linear Equation in Two Variables

Category : 9th Class

### Introduction

Previously we have studied about a linear polynomial in two variable. The general form of linear polynomial in two variable is $ax+by+c$, where a, $b\ne 0$. In this chapter we will study about linear equation in two variable.

### Linear Equation in Two Variables

Suppose $p(x)=ax+by+c$ is a linear polynomial in two variable where a, $b\ne 0$. Then $p(x)=0$ is a linear equation in two variable i.e. $ax+by+c=0$, where a, $b\ne 0$ is a linear equation in two variable.

Solution of a Linear Equation

$x=p$and$y=q$ is called the solution of a linear equation $ax+by+c=0$ if $ap+bq+c=0$

Which one of the following equations is not a linear equation in two variable?

(a) $4x+\frac{7}{2}y=4$

(b) $\frac{4}{x}+\frac{7}{y}=4$

(c) $\frac{x+3}{x-3}=4$

(d) $3x+7y+87=0$

(e) None of these

Explanation:

In $\frac{4}{x}+\frac{7}{y}=4$

The power of variable x and y is -1. Therefore, it is not a linear equation.

Graph a Linear Equation in Two Variable

The general form of linear equation in two variable is $ax+by+c=0$

$\Rightarrow$$by=ax-c$ $\Rightarrow$$y=\left( \frac{-a}{b} \right)x-\frac{c}{a}$

It is the form

$y=mx+c$ Represents a line where $m=\left( \frac{-a}{b} \right)$ and $c=\frac{c}{a}$ and m is known as the slope of this line. That is why, we can say that the graph of a linear equation represented by a line.

The slope of line $4x+\text{3y}-\text{4}=0$ is______

(a) $\frac{3}{4}$

(b) $\frac{-4}{3}$

(c) $\frac{4}{3}$

(d) $\frac{-3}{4}$

(e) None of these

Explanation:

$4x+3y-4=0$

$\Rightarrow$$3y=-4x+4$ $\Rightarrow$$y=\left( \frac{-4}{3} \right)x+4$

Here, $m=\frac{-4}{3}$

Graph of $\mathbf{ax+by+c=0}$

The following steps are followed to draw a graph:

Step 1:   Express $x$ in terms of y or y in terms of $x$.

Step 2:   Select at least three values of y or $x$ and find the corresponding values of$x$ or y respectively, which satisfies the given equation, write these values of $x$ and y in the form of table.

 $x$ $y$

Step 3:   Plot ordered pair ($x$, y) from the table on a graph paper.

Step 4:   Join these points by a straight line. Note: Every point on the line is a solution of linear equation in two variable i.e. there are infinite number of solution of a linear equation.

Draw the graph of $4x-y+3=0$.

Solution:

Step 1:   Here it is easy to write y in terms of$x$ $\Rightarrow$ $y=4x+3$.

Step 2:   $y=4x+3$

if $x=0$      $\Rightarrow$               $y=3$

$\Rightarrow$$x=-1$               $\Rightarrow$               $y=-1$

 $x$ 0 1 -1 $y$ 3 7 -1

Step 3 and Step 4:

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