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

         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  

Answer: (b)  

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  

Answer: (b)  

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.  

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 \[x=0\]      \[\Rightarrow \]               \[y=3\]  

\[\Rightarrow \]\[x=-1\]               \[\Rightarrow \]               \[y=-1\]  

Step 3 and Step 4: