Simultaneous Equations Graphically

How to solve a pair of simultaneous equations graphically?

In order to solve a pair of simultaneous equations graphically, we first draw the graph of the two equations simultaneously. We get two straight lines intersecting each other at a common point. This common point intersection of two lines gives the solution of the pair of simultaneous equations.


How to find the solution of simultaneous linear equations by graphing method?

Suppose the given system of linear equations is

p\(_{1}\)x + q\(_{1}\)y + r\(_{1}\) = 0 ------------- Equation (1)

p\(_{2}\)x + q\(_{2}\)y + r\(_{2}\) = 0 ------------- Equation (2)

Now draw the both given linear equations on a graph paper. Graph of linear equation is always a straight line.

Let the line L\(_{1}\) represent the graph of p\(_{1}\)x + q\(_{1}\)y + r\(_{1}\) = 0 and L\(_{2}\) represent the graph of p\(_{2}\)x + q\(_{2}\)y + r\(_{2}\) = 0.

Then by solving the simultaneous equations graphically we can get three possible solutions.


1. When the two lines L\(_{1}\) and L\(_{2}\) intersect at a single point:

Then the two lines cross at a single point P (h, k), as shown in the graph.

Simultaneous Equations Graphically

Therefore, x = h and y = k is the unique solution of the given two equations.

This system is called independent.


Solved examples of a pairs of simultaneous equations graphically where the two lines intersect at a single point:

Solve graphically the system if linear equation x + y = 5 and x – y = 3.

Solution:            

On a graph paper to solve simultaneous equations graphically, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis respectively.

x + y = 5 ------------ Equation (1)

Now convert the given equation in the form of y = mx + b

x + y = 5

⇒ x - x + y = - x + 5

⇒ y = - x + 5            

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = -x + 5.

When the value of x = 1 then y = -1 + 5 = 4  

When the value of x = 2 then y = -2 + 5 = 3

When the value of x = 3 then y = -3 + 5 = 2

When the value of x = 4 then y = -4 + 5 = 1

Arrange these value of the linear equation y = -x + 5 in the table 1.    

Linear Equation Table

Table 1

Now plot the points of the equation x + y = 5; A (1, 4), B (2, 3), C (3, 2), D (4, 1) on the graph paper.                            

Join the points of A, B, C and D; to get the graph line AD.

Thus line AD is the graph of x + y = 5.

x – y = 3 ------------ Equation (2)

Now convert the given equation in the form of y = mx + b

x – y = 3

⇒ x – x – y =  – x + 3

⇒ -y = -x + 3

⇒ y = x – 3

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = x - 3.

When the value of x = 1 then y = 1 – 3 = -2

When the value of x = 2 then y = 2 – 3 = -1

When the value of x = 3 then y = 3 – 3 = 0

When the value of x = 4 then y = 4 – 3 = 1

Arrange these value of the linear equation y = x - 3 in the table 2.  

Linear Equation Table 2

Table 2

Now plot the points of the equation x – y = 3; P (1, -2), Q (2, -1), R (3, 0), S (4, 1) on the graph paper.                          

Join the points of P, Q, R and S; to get the graph line PS.

Thus line PS is the graph of x – y = 3.

Two Lines Intersect at a Single Point

We get two straight lines intersecting each other at (4, 1).

Therefore, x = 4 and y = 1 is the solution of the given system of equation.


2. When the two lines L\(_{1}\) and L\(_{2}\) are coincident:

Then the two equations actually be the same line, as shown in the graph.

Two Lines are Coincident

Therefore, the given two equations have infinitely many solutions.

This system is called dependent.


Solved examples of a pairs of simultaneous equations graphically where the two lines coincidentally be the same line:

Solve graphically the system if linear equation 3x - y = 2 and 9x – 3y = 6.

Solution:            

On a graph paper to solve simultaneous equations graphically, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis respectively.

3x - y = 2 ------------ Equation (1)

Now convert the given equation in the form of y = mx + b

y = 3x - 2             

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = 3x - 2.

When the value of x = -1 then y = 3(-1) - 2 = -5

When the value of x = 0 then y = 3(0) - 2 = -2

When the value of x = 2 then y = 3(2) - 2 = 4 

Arrange these value of the linear equation y = 3x - 2 in the table 1.   

Trial and Error Method

Table 1

Now plot the points of the equation 3x - y = 2; A (-1, -5), B (0, -2), C (2, 4) on the graph paper.                        

Join the points of A, B and C; to get the graph line AC.

Thus line AC is the graph of 3x - y = 2.

9x – 3y = 6 ------------ Equation (2)

Now convert the given equation in the form of y = mx + b

y = (9x – 6)/3

⇒ y = 9x/3 – 6/3

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = 9x/3 – 6/3.

When the value of x = -2 then y = -8

When the value of x = 1 then y = 1

When the value of x = 2 then y = 4

Arrange these value of the linear equation y = 9x/3 – 6/3 in the table 2.  

Points of the Equation

Table 2

Now plot the points of the equation 9x – 3y = 6; P (1, -2), Q (2, -1), R (3, 0) on the graph paper.                      

Join the points of P, Q and R; to get the graph line PR.

Thus line PR is the graph of 9x – 3y = 6.

Two Lines Coincidentally be the Same Line


We find that the two straight lines AC and PR coincides each other.

Therefore, the given system of equations has an infinite number of solutions.


3. When the two lines L\(_{1}\) and L\(_{2}\) are parallel:

Then the two equations have no common solutions, as shown in the graph.

Two Lines are Parallel

This system is called inconsistent.


Solved examples of a pairs of simultaneous equations  graphically when the two lines are parallel:

Solve graphically the system if linear equation 2x - 3y = 5 and 6y – 4x = 3.

Solution:            

On a graph paper to solve simultaneous equations graphically, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis respectively.

2x - 3y = 5 ------------ Equation (1)

Now convert the given equation in the form of y = mx + b

2x - 3y = 5

⇒ 3y = (2x – 5)

⇒ y = (2x – 5)/3

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = (2x – 5)/3

When the value of x = -2 then y = -3                                                       

When the value of x = 1 then y = -1                                                        

When the value of x = 4 then y = 1                                                          

Arrange these value of the linear equation y = (2x – 5)/3 in the table 1.

Plot the Points of the Equation

Table 1

Now plot the points of the equation 2x - 3y = 5; A (-2, -3), B (1, -1) and C (4, 1) on the graph paper.

Join the points of A, B and C; to get the graph line AC.

Thus line AC is the graph of 2x - 3y = 5.

6y – 4x = 3 ------------ Equation (2)

Now convert the given equation in the form of y = mx + b

6y – 4x = 3

⇒ 6y = (3 + 4x)

⇒ y = (3 + 4x)/6

Now we will apply the trial and error method to find 3 pairs of values of (x, y) which satisfy the given equation y = (3 + 4x)/6.

When the value of x = -3 then y = -3/2

When the value of x = 0 then y = 1/2

When the value of x = 3 then y = 5/2

Arrange these value of the linear equation y = (3 + 4x)/6 in the table 2.

System of Equations Points

Table 2

Now plot the points of the equation 6y – 4x = 3; P (-3, -3/2), Q (0, 1/2), R (3, 5/2) on the graph paper.     

Join the points of P, Q and R; to get the graph line PR.

Thus line PR is the graph of 6y – 4x = 3.

The Two Lines are Parallel

We get from the graph that two straight lines are parallel to each other.

Therefore, the given system of equations has no solutions.


Thus we conclude that to solve a pair of simultaneous equations graphically then the possible outcomes can be:

(i) a unique solution if the graph lines intersect at a point.

(ii) infinitely many solutions if the two graph lines coincides.

(iii) no solution if the two graph lines are parallel.

Related Concepts:

Coordinate Graph

Ordered pair of a Coordinate System

Plot Ordered Pairs

Coordinates of a Point

All Four Quadrants

Signs of Coordinates

Find the Coordinates of a Point

Coordinates of a Point in a Plane

Plot Points on Co-ordinate Graph

Graph of Linear Equation

Simultaneous Equations Graphically

Graphs of Simple Function

Graph of Perimeter vs. Length of the Side of a Square

Graph of Area vs. Side of a Square

Graph of Simple Interest vs. Number of Years

Graph of Distance vs. Time






7th Grade Math Problems

8th Grade Math Practice

From Simultaneous Equations Graphically to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Subtracting Integers | Subtraction of Integers |Fundamental Operations

    Jun 13, 24 02:51 AM

    Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

    Read More

  2. Properties of Subtracting Integers | Subtraction of Integers |Examples

    Jun 13, 24 02:28 AM

    The properties of subtracting integers are explained here along with the examples. 1. The difference (subtraction) of any two integers is always an integer. Examples: (a) (+7) – (+4) = 7 - 4 = 3

    Read More

  3. Math Only Math | Learn Math Step-by-Step | Worksheet | Videos | Games

    Jun 13, 24 12:11 AM

    Presenting math-only-math to kids, students and children. Mathematical ideas have been explained in the simplest possible way. Here you will have plenty of math help and lots of fun while learning.

    Read More

  4. Addition of Integers | Adding Integers on a Number Line | Examples

    Jun 12, 24 01:11 PM

    Addition of Integers
    We will learn addition of integers using number line. We know that counting forward means addition. When we add positive integers, we move to the right on the number line. For example to add +2 and +4…

    Read More

  5. Worksheet on Adding Integers | Integers Worksheets | Answers |Addition

    Jun 11, 24 07:15 PM

    Worksheet on Adding Integers
    Practice the questions given in the worksheet on adding integers. We know that the sum of any two integers is always an integer. I. Add the following integers:

    Read More