An equation in the form of \(ax+b=0\), where \(a, b\) are real numbers, and \(a≠0\) is called a linear equation in one variable. And every linear equation in one variable has a unique solution. Now, an equation of form \(ax+by+c=0,\) where \(a, b\) and \(c\) are real numbers and \(a\) and \(b\) is non-zero, is called a general linear equation in the two variables \(x\) and \(y.\) \(x=\alpha\) and \(y=\beta\) is a solution of the linear equation \(ax+by+c=0\) if and only if \(a \alpha+b \beta+c=0\), where \(\alpha\) and \(\beta\) are real numbers. In this article, we will learn Graphical Method of Solution of a Pair of Linear Equations. Read on to learn more about the method.

Graphical Method of Finding Solution of a Pair of Linear Equations

To solve graphically a system of two simultaneous linear equations in two variables \(x\) and \(y,\) we should proceed as shown below.

1. Draw a graph for each of the given linear equations.
2. Find the coordinates of the point of intersection of the two lines drawn.
3. The coordinates of the point of intersection of the two lines will be the common solution of the given equations.
4. Lastly, write the values of \(x\) and \(y.\)
5. Later, check the above solution by substituting the values of \(x\) and \(y\) obtained in both the given equations.

Depending on the solution of the equations, the plotting of two lines on a graph can be summarised as:

1. If the two equations have a unique common solution, then the equations are called consistent and independent. In this case, the lines have one and only one point in common.
2. If the two equations have several common solutions, then the equations are called consistent and dependent. In this case, the two lines will be coincident.
3. And, if the two equations have no common solution, then the equations are called inconsistent. In this case, the two lines will be parallel.

Applications of Graphical Method of Solution of a Pair of Linear Equations 

The graphical representation of the equation can be used to find the area of the triangle formed. It can also be used to find the vertices of the triangle whose equations are given. Let us understand its application.

Example 1: Solve the equations \(3y=4x\) and \(2x+3y=18\) graphically. Also, find the ratio of the areas of the triangles formed by these lines and the coordinates axes.

Solution: The given equations are: \(3y=4x\) and \(2x+3y=18\)
Thus, the equation can be written as \(y=\frac{4}{3} x \ldots (i)\)
And \(y=\frac{18-2 x}{3} \ldots (ii)\)

Pair of Linear Equations in Two Variables

Table of values for the equation (i)

\(x\)	\(0\)	\(3\)	\(6\)
\(y\)	\(0\)	\(4\)	\(8\)

Table of values for the equation (ii)

\(x\)	\(0\)	\(3\)	\(6\)
\(y\)	\(6\)	\(4\)	\(2\)

Select coordinate axes and take \(1 \mathrm{~cm}=1\) unit on both the axes. Plot the above-obtained points \((0, 0), (3,4),\) and \((6,8)\) on a graph paper. Connect any two points by a straight line.

Plot the points \((0, 6), (3, 4)\), and \((6, 2)\) on the same graph. Connect any two points by a straight line. The graphs of both the straight line are shown in the figure given below.

The line intersects at the point \(P(3, 4).\) Therefore, the solution of the given equations is \(x=3\) and \(y=4.\)

Now, as we can see from the figure that the line \(3y=4x\) passes from the origin, and line \(2x+3y=18\) meets the \(x-\) axis at the point \(A(9, 0)\) and the \(y\)-axis at the point \(B(0, 6)\). There are \(2\) triangles formed, i.e., \(\Delta POA\) and \(\Delta OPB\) formed by these lines and the coordinate axes. 

From \(P,\) draw a perpendicular to \(OA,\) and \(PN\) perpendicular to \(OB.\)

Area of \(\Delta POA = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times OA \times MP\)
\(=\frac{1}{2} \times 9 \times 4=18\) square units.

Ans, area of \(\Delta OPB = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times OB \times NP\)
\(=\frac{1}{2} \times 6 \times 3=9\) square units.

Therefore, \(\frac{{{\rm{ Area \,of\, }}\Delta POA}}{{{\rm{ Area\, of\, }}\Delta OPB}} = \frac{{18}}{9} = \frac{2}{1}\)

Hence, the \({\rm{area}}\,{\rm{of}}\,\Delta POA:{\rm{area}}\,{\rm{of}}\,\Delta OPB = 2:1.\)

Solved Examples – Graphical Method of Solution of a Pair of Linear Equations

Q.1. Draw the pair of given lines graphically.
y=2-x, and  2y=4-2x
Ans: Let \(y=2-x… (i)\) and \(2y=4-2x… (ii)\)
Table for equation (i)

\(x\)	\(0\)	\(2\)
\(y\)	\(2\)	\(0\)

Table for equation (ii)

\(x\)	\(0\)	\(1\)
\(y\)	\(2\)	\(1\)

The graphs of both the straight line are shown in the figure given below.

From the linear equation graph, it is clear that the given two equations coincide.

Q.2. Draw the pair of given lines graphically.
y=2-x, and y=3-x
Ans: Let \(y=2-x… (i)\) and \(y=3-x… (ii)\)
Table for equation (i)

\(x\)	\(0\)	\(2\)
\(y\)	\(2\)	\(0\)

Table for equation (ii)

\(x\)	\(0\)	\(3\)
\(y\)	\(3\)	\(0\)

The graphs of both the straight line are shown in the figure given below.

From the linear equation graph, it is clear that they are parallel, and they will not intersect anywhere. 

Q.3. Solve the following system of equations graphically:
4x=y+5, 5y=7+4x
Ans: The given equations can be written as;
\(y=4x-5… (i)\)
\(y = \frac{1}{5}\left( {4x + 7} \right)…(ii)\)
Table for equation (i)

\(x\)	\(1\)	\(0\)	\(3\)
\(y\)	\(-1\)	\(-5\)	\(7\)

Table for equation (ii)

\(x\)	\(-3\)	\( – \frac{1}{2}\)	\(2\)
\(y\)	\(-1\)	\(1\)	\(3\)

Select coordinate axes and take \(1 \mathrm{~cm}=1\) unit on both the axes. Plot the points \((1,-1), (0,-5),\) and \((3,7)\) on a graph paper. Connect any two points by a straight line. Plot the points \((-3,-1),\left(-\frac{1}{2}, 1\right)\) and \((2, 3)\) on the same graph. Connect any two points by a straight line. The graphs of both the straight line are shown in the figure given below.

The line intersects at point \(P(2, 3).\) Therefore, the solution of the given equations is \(x=2, y=3.\)

Learn About Solution of a Linear Equation

Q.4. Find graphically the triangle’s vertices whose sides have equations 2y=8+x, 5y=14+x, and y=2x+1, respectively. 
Ans: The given equations are
\(2y=8+x… (i)\)              \(5y=14+x… (ii)\)            \( y=2x+1… (iii)\)
Table of values for the equation (i)

\(x\)	\(0\)	\(2\)	\(-2\)
\(y\)	\(4\)	\(5\)	\(3\)

Table of values for the equation (ii)

\(x\)	\(-4\)	\(1\)	\(6\)
\(y\)	\(2\)	\(3\)	\(4\)

Table of values for the equation (iii)

\(x\)	\(0\)	\(1\)	\(2\)
\(y\)	\(1\)	\(3\)	\(5\)

Take \(1 \mathrm{~cm}=1\) unit. Plot the points \((0, 4), (2, 5),\) and \((-2, 3)\) on a graph paper. Connect any two points by a straight line.
Plot the points \((-4, 2), (1, 3),\) and \((6, 4)\) on the same graph and connect any two points by a straight line. 
Plot the points \((0, 1), (1, 3),\) and \((2, 5)\) on the same graph and connect any two points by a straight line. The graphs of both the straight line are shown in the figure given below.

Q.5. A triangle is formed by the lines x=2y+3, 3x=2y-7 and y+1=0. Find the area of the triangle graphically.
Ans: The given equations can be written as \(y=-\frac{1}{2}(x-3) \ldots (i)\)
\(y=-\frac{1}{2}(3 x+7) \ldots (ii)\)
\(y=-1 \ldots (iii)\)
Table of values for the equation (i)

\(x\)	\(3\)	\(1\)	\(-1\)
\(y\)	\(0\)	\(1\)	\(2\)

Table of values for the equation (ii)

\(x\)	\(-1\)	\(-2\)	\(-3\)
\(y\)	\(2\)	\(\frac{1}{2}\)	\(-1\)

Table of values for the equation (iii)

\(x\)	\(0\)	\(2\)	\(4\)
\(y\)	\(-1\)	\(-1\)	\(-1\)

Select coordinate axes and take \(1 \mathrm{~cm}=1\) unit. Plot the points \((3, 0), (1, 1),\) and \((-1, 2)\) on a graph paper. Connect any two points by a straight line.

Plot the points \((-1,2),\left(-2, \frac{1}{2}\right)\) and \((-3, -1)\) on the same graph and connect any two points by a straight line. 

Plot the points \((0, -1), (2, -1),\) and \((4, -1)\) on the same graph and connect any two points by a straight line. The graphs of both the straight line are shown in the figure given below.

Summary

In this article, we first learnt the definition of a linear equation in \(2\) variables. Then, we learned to find the coordinates and then learnt to plot the points on the graph. In addition to this, we also learnt the applications of the graphical solution of the linear pair of equations. An equation of form (ax+by+c=0,) where (a, b) and (c) are real numbers and (a) and (b) is non-zero, is called a general linear equation in the two variables (x) and (y.) To find the solution of the given equation, plot the points on the graph and find whether the lines are intersecting, parallel or coinciding.

(i) If the two equations have a unique common solution, then the equations are called consistent and independent. In this case, the lines have one and only one point in common.
(ii) If the two equations have several common solutions, then the equations are called consistent and dependent. In this case, the two lines will be coincident.
(iii) And, if the two equations have no common solution, then the equations are called inconsistent. In this case, the two lines will be parallel.

FAQs on Graphical Method of Solution of a Pair of Linear Equations

The most commonly asked questions about Group 13 elements are answered here:

Q.1. How do you solve linear equations by graphical method? Ans: To solve graphically, a system of two simultaneous linear equations in two variables \(x\) and \(y\) proceed as shown below. (i) Draw a graph for each of the given linear equations. (ii) Find the coordinates of the point of intersection of the two lines drawn. (iii) The coordinates of the point of intersection of the two lines will be the common solution of the given equations. (iv) Lastly, write the values of \(x\) and \(y.\) (v) Later, check the above solution by substituting the values of \(x\) and \(y\) obtained in both the given equations.

Q.2. How do you solve two linear equations graphically? Ans: The basic approach is to represent them as straight lines on the graph and find the points of intersection.

Q.3. How do you find the graphical method? Ans: To find the solution of the given equation, plot the points on the graph and find whether the lines are intersecting, parallel or coinciding.

Q.4. What is a linear equation? Ans: An equation of form \(ax+by+c=0,\) where \(a, b\) and \(c\) are real numbers and \(a\) and \(b\) is non-zero, is called a general linear equation in the two variables \(x\) and \(y.\)

Q.5. What are the graphical conditions for the pair of linear equations which has no solution? Ans: Depending on the solution of the equations, the plotting of two lines on a graph can be summarised as (i) If the two equations have a unique common solution, then the equations are called consistent and independent. In this case, the lines have one and only one point in common. (ii) If the two equations have several common solutions, then the equations are called consistent and dependent. In this case, the two lines will be coincident. (iii) And, if the two equations have no common solution, then the equations are called inconsistent. In this case, the two lines will be parallel.

Now you are provided with all the necessary information on the solution of a pair of linear equations using the graphical method and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.