Laravel

Chapters For Class X- CBSE


Graphical Method of Solution of a Pair of Linear Equations
Behaviour of lines representing a pair of linear equations in two variables.


Consider a pair of lines represented by the equations : 

      ${a_1x + b_1y + c_1  =  0}$ and      
           ${a_2x + b_2y + c_2  =  0}$

Hence, ${a_1x + b_1y + c_1  =  0}$ can be written as 
             $y=-{a_1/b_1}x-c_1/b_1$  .....................(1)
and  ${a_2x + b_2y + c_2  =  0}$ can be written as 
             $y=-{a_2/b_2}x-c_2/b_2$  .....................(2)

We know that the standard equation of straight line is $y=mx+c$
where $m$ is the angle (slope) which the line make with $x-$ axis and $c$ is the intercept on $y-$ axis.
1)If the lines are parallel.
If the lines are parallel , then the slope of two lines are same but the intercept on $y-$ axis is different,i.e
$a_1/b_1=a_2/b_2$ and $c_1/b_1≠c_2/b_2$
i.e    $a_1/a_2=b_1/b_2$ and $b_1/b_2≠c_1/c_2$
Hence,If the lines are parallel,the pair of linear equations has no solution, and is called an inconsistent pair of linear equations.

 
   ${a_1/a_2=b_1/b_2≠c_1/c_2}$ 

    e.g.    ${x + 2y – 4 = 0}$ and ${2x + 4y – 12 =0 }$
2) If the lines are intersecting
If the lines are intersecting , then the slope of the lines are never equal , but the intercept on $y-$ axis can be equal.
i.e   $a_1/b_1≠a_2/b_2$
i.e $a_1/a_2≠b_1/b_2$
Hence, if the lines are intersecting,then the pair of linear equations has one solution,and is called a consistent pair of linear equations. 

  
  ${a_1/a_2≠b_1/b_2}$ 

    e.g.   ${x – 2y = 0}$  and ${3x + 4y – 20 = 0}$
3) If the lines coincides (or are equivalent).
If the lines coincides, then the slope and intercept on $y-$ axis are same.
i.e $a_1/b_1=a_2/b_2$ and $c_1/b_1=c_2/b_2$
i.e $a_1/a_2=b_1/b_2$ and $b_1/b_2=c_1/c_2$
i.e $a_1/a_2=b_1/b_2=c_1/c_2$
Hence,the pair of linear equations has infinitely many distinct common solutions. Such a pair is called a dependent pair of linear equations in two variables. Note that a dependent pair of linear equations is always consistent. 
  
 ${a_1/a_2=b_1/b_2=c_1/c_2}$ 

    e.g.   ${ 2x + 3y – 9 = 0}$  and ${4x + 6y – 18 = 0}$
Example: Check graphically whether the pair of equations 
		${x + 3y =  6 }$      (1)  and           
		${2x – 3y = 12}$      (2) 
is consistent. If so, solve them graphically.

Solution: Let us draw the graphs of the Equations (1) and (2). 

 

  

		 
$x$0 6 

		 
y= (6-x)/3 2 0
		 
		 

		 
		 

		 
$x$0 6 

		 
y=(12+2x)/3 -4  0
We observe that there is a point B (6, 0) common to both the lines AB and PQ. So, the solution of the pair of linear equations is $x = 6$ and $y = 0$, i.e., the given pair of equations is consistent.

Example: Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions: 
		${5x – 8y + 1 =  0 }$                             (1)
                                                        
		${3x-24/5y+3/5=0}$                                (2)


Solution : Multiplying Equation (2) by $5/3$ , we get
 
   ${3(5/3)x-24/5({5/3})y+3/5(5/3)=0}$
   
   so we get,
   ${5x – 8y + 1 =  0}$
But, this is the same as Equation (1). Hence the lines represented by Equations (1) and (2) are coincident. Therefore, Equations (1) and (2) have infinitely many solutions.


Example: Champa went to a ‘Sale’ to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased”. Help her friends to find how many pants and skirts Champa bought.
Solution : Let us denote the number of pants by $x$ and the number of skirts by $y$. Then the equations formed are : 
 
				${y =  2x – 2}$        (1)
				and ${y =  4x – 4}$     (2)

 Draw  the  graphs  of Equations (1) and (2) by finding two solutions for each of the equations. 
     


  

		 
$x$0 1 

		 
y= 2x – 2 -2 0
		 
		 

		 
		 

		 
$x$0 1 

		 
y= 4x – 4 -4  0 
		 
 

                                               
The two lines intersect at the point (1, 0). So, $x = 1$, $y = 0$ is the required solution of the pair of linear 
equations, i.e., the number of pants she purchased is 1 and she did not buy any skirt.