रैखिक समीकरण युग्म का ग्राफीय विधि से हल

जब गणितीय संक्रियाओं के साथ चरों और अचरों के गणितीय व्यंजक उच्चतम घात एक का समीकरण बनाते हैं, तो इसे एक रैखिक समीकरण कहा जाता है। रैखिक समीकरण चरों के बीच एक बीजीय समीकरण है जो आलेख पर अंकित करने पर एक सीधी रेखा देता है। एक चर का एक रैखिक समीकरण इस प्रकार का होता है $ax+b=0$ जहां $x$ चर है। दो चरों के रैखिक समीकरण इस रूप के होते हैं $ax+by+c=0$ जहाँ $x$ और $y$ दो चर हैं और $c$ स्थिरांक है। रैखिक समीकरणों की एक युग्म को दो मूल विधियों का उपयोग करके हल किया जा सकता है और दर्शाया जा सकता है: आलेखीय विधि और बीजगणितीय विधि। इस पाठ में, हम आलेखीय विधि का उपयोग करके दो रैखिक समीकरणों की एक प्रणाली को हल करने की विधि को ज्ञात करेंगे।

Solving Pair of Linear Equation Graphically

Every linear equation consists of variables. Linear equations are of the first order and they may involve one or two variables. When it comes to solving linear equations using graphical method the basic approach is to represent them as straight lines on a graph and find the points of intersection, if any. We can obtain at least two solutions easily by substituting the values for $x$, finding the $x$ and $y$ intercepts and plotting them geometrically on the graph. Let us have a look at the standard form of a pair of linear equations here.

$a_{1}x+b_{1}y+c_{1}=0....(1)$

$a_{2}x+b_{2}y+c_{2}=0....(2)$

Solution for the equations varies according to the position of the lines.

Types of Solutions

• Consistent : The pair of equations is said to be consistent, if the two lines are intersecting at the same point, then the point gives unique solution for both the equations.
• Dependent : The pair of equations is said to be dependent, if the two lines coincide, then in this case there are infinitely many solutions. Each and every point on a line becomes a solution.
• Inconsistent : The pair of equations is said to be inconsistent, if the two lines are parallel, then in this case there is no solution.

Consider the following three pairs of equations.

(i) $x-2y=0$and $3x+4y-20=0$ (The lines intersect)

(ii) $2x+3y-9=0$ and $4x+6y-18=0$ (The lines coincide)

(iii) $x+2y-4=0$ and $2x+4y-12=0$ (The lines are parallel)

Let us write down, and compare, the values of ${\frac {a_{1}}{a_{2}}},{\frac {b_{1}}{b_{2}}},{\frac {c_{1}}{c_{2}}}$

in all the three examples. Here $a_{1},b_{1},c_{1}$and $a_{2},b_{2},c_{2}$ denote the coefficients of equations given in in the general form $(1)$and (2)

Sl.No.	Pair of Lines	${\frac {a_{1}}{a_{2}}}$	${\frac {b_{1}}{b_{2}}}$	${\frac {c_{1}}{c_{2}}}$	Compare the ratios	Graphical representation	Algebraic Interpretation
1	$x+3y-6=0$ $2x-3y-12=0$	${\frac {1}{2}}$	${\frac {3}{-3}}$	${\frac {-6}{-12}}$	${\frac {a_{1}}{a_{2}}}\neq {\frac {b_{1}}{b_{2}}}$	Intersecting lines	Exactly one solution (unique)
2	$2x+3y-9=0$ $4x+6y-18=0$	${\frac {2}{4}}$	${\frac {3}{6}}$	${\frac {-9}{-18}}$	${\frac {a_{1}}{a_{2}}}={\frac {b_{1}}{b_{2}}}={\frac {c_{1}}{c_{2}}}$	Coincident lines	Infinitely many solutions
3	$x+2y-4=0$ $2x+4y-12=0$	${\frac {1}{2}}$	${\frac {2}{4}}$	${\frac {-4}{-12}}$	${\frac {a_{1}}{a_{2}}}={\frac {b_{1}}{b_{2}}}\neq {\frac {c_{1}}{c_{2}}}$	Parallel lines	No Solution

From the table above, if the lines represented by the equation

$a_{1}x+b_{1}y+c_{1}=0....(1)$ and $a_{2}x+b_{2}y+c_{2}=0....(2)$ are

• intersecting , then ${\frac {a_{1}}{a_{2}}}\neq {\frac {b_{1}}{b_{2}}}$
• coincident , then ${\frac {a_{1}}{a_{2}}}={\frac {b_{1}}{b_{2}}}={\frac {c_{1}}{c_{2}}}$
• parallel , then ${\frac {a_{1}}{a_{2}}}={\frac {b_{1}}{b_{2}}}\neq {\frac {c_{1}}{c_{2}}}$

Examples

1. Check graphically whether the pair of equations

$x+3y-6=0$ $2x-3y-12=0$

is consistent. If so, solve them graphically.

Solution:

$x$	$0$	$6$
$y={\frac {6-x}{3}}$	$2$	$0$

$x$	$0$	$3$
$y={\frac {2x-12}{3}}$	$-4$	$-2$

Plot the points on the graph paper

• $(0,2)$, $(6,0)$ and join the points to form the lines
• $(0,-4)$ $(3,-2)$ and join the points to form the lines as shown in Fig. 1.

Fig.1

We observe that there is a point at $(6,0)$ common to both the lines . So, the solution of the pair of linear equations is $x=6$ and $y=0$, i.e., the given pair of equations

is consistent.

2. Check graphically whether the pair of equations

$2x+3y-9=0$ $4x+6y-18=0$

has infinitely many solutions. If so, solve them graphically.

Solution:

$x$	$3$	$6$
$y={\frac {9-2x}{3}}$	$1$	$-1$

$x$	$3$	$6$
$y={\frac {18-4x}{6}}$	$1$	$-1$

Plot the points on the graph paper

• $(3,1)$, $(6,-1)$ and join the points to form the lines
• $(3,1)$ $(6,-1)$ and join the points to form the lines as shown in Fig. 2.

Fig. 2

We observe that each and every point on a line becomes a solution. So, the solution of the pair of linear equations has infinitely many solutions.

3. Check graphically whether the pair of equations

$x+2y-4=0$ $2x+4y-12=0$

has no solution, If so, solve them graphically.

Solution:

$x$	$0$	$2$
$y={\frac {4-x}{2}}$	$2$	$1$

$x$	$0$	$2$
$y={\frac {12-2x}{4}}$	$3$	$2$

Plot the points on the graph paper

• $(0,2)$, $(2,1)$ and join the points to form the lines
• $(0,3)$ $(2,2)$ and join the points to form the lines as shown in Fig. 3

Fig. 3

We observe that lines are not crossing and are parallel to each other . So, the pair of linear equations has no solution.