In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Maths > Pair of Linear Equations in Two Variables > Consistency of Pair of Linear Equations in Two Variables
Pair of Linear Equations in Two Variables

Consistency of Pair of Linear Equations in Two Variables

A pair of linear equations in two variables have the same set of variables across both the equations. These equations are solved simultaneously to arrive at a solution. In this article, we will look at the various types of solutions of equations in two variables.

Suggested Videos

Play
Play
Play
Arrow
Arrow
ArrowArrow
Introduction to Linear Equations in Two Variables
Solution to Question ID 626878
Consistency of Pair of Linear Equations
Slider

 

Types of Linear Equations in Two Variables

Solutions of linear equations in two variables can be of three types

  1. Single Solution
  2. Infinite Solutions
  3. No Solution

Understanding these types will help us in solving linear equations in two variables effectively. We will look at each of them in details.

Type 1: A single solution of a pair of linear equations in two variables

Consider the following pair of linear equations in two variables,

  • x – 2y = 0
  • 3x + 4y = 20

The solution of this pair would be a pair (x, y). Let’s find the solution, geometrically. The tables for these equations are:

x 0 2
y = (1/2)x 0 1

 

x 0 4
y = (20 – 3x)/4 5 2

Now, take a graph paper and plot the following points:

  • A(0, 0)
  • B(2, 1)
  • P(0, 5)
  • Q(4, 2)

Next, draw the lines AB and PQ as shown below.

equations in two variables

From the figure above, you can see that the two lines intersect at the point Q (4, 2). Therefore, point Q lies on the lines represented by both the equations, x – 2y = 0 and 3x + 4y = 20. Hence, (4, 2) is the solution of this pair of equations in two variables. Let’s verify it algebraically:

  • x – 2y = 4 – 2(2) = 4 – 4 = 0 = RHS
  • 3x + 4y = 3(4) + 4(2) = 12 + 8 = 20 = RHS

Further, from the graph, you can see that point Q is the only common point between the two lines. Hence, this pair of equations has a single solution.

Browse more Topics under Pair Of Linear Equations In Two Variables

Download NCERT Solutions for Class 10 Mathematics

Type 2: Infinite solutions of a pair of linear equations in two variables

Consider the following pair of linear equations in two variables,

  • 2x + 3y = 9
  • 4x + 6y = 18

The solution of this pair would be a pair (x, y). Let’s find the solution, geometrically. The tables for these equations are:

x 0 4.5
y = (9 – 2x)/3 3 0

 

x 0 3
y = (18 – 4x)/6 3 1

Now, take a graph paper and plot the following points:

  • A(0, 3)
  • B(4.5, 0)
  • P(3, 1)

Next, draw the lines AB and AP as shown below.

equations in two variables

From the figure above, you can see that the two lines coincide. Therefore, there is no point of intersection between these two lines. Every point on the line represented by 2x + 3y = 9 is present on the line represented by 4x + 6y = 18. Hence, this pair of equations has an infinite number of solutions.

Download Pair of Linear Equations in Two Variables Cheat Sheet by clicking on the button below
Pair of Linear Equations in Two Variables

Type 3: No solution of a pair of linear equations in two variables

Consider the following pair of linear equations in two variables,

  • x + 2y = 4
  • 2x + 4y = 12

The solution of this pair would be a pair (x, y). Let’s find the solution, geometrically. The tables for these equations are:

x 0 4
y = (4 – x)/2 2 0

 

x 0 6
y = (12 – 2x)/4 3 0

Now, take a graph paper and plot the following points:

  • A(0, 2)
  • B(4, 0)
  • P(0, 3)
  • Q(6, 0)

Next, draw the lines AB and PQ as shown below.

equations in two variables

From the figure above, you can see that the two lines are parallel to each other. Therefore, these lines don’t intersect at all. Hence, this pair of equations has no solution.

Consistency of a Pair of Linear Equations in Two Variables

From the three examples above, we define the following terms

  • Inconsistent pair of linear equations: A pair of linear equations which has no solution. (Type 3 explained above)
  • Consistent pair of linear equations: A pair of linear equations which has a solution. (Type 1 and 2 explained above)
    • Independent pair of linear equations – If the pair of equations has only one solution.
    • Dependent pair of linear equations – If the pair of equations has infinite solutions.

To summarise, the lines represented by

  • (x – 2y = 0) and (3x + 4y = 20) intersect each other.
  • 2x + 3y = 9 and 4x + 6y = 18 coincide with each other.
  • ‘x + 2y = 4’ and ‘2x + 4y = 12’ are parallel to each other.

In General Form

If a1, b1, c1, a2, b2 and c2 are the coefficients of the equations in general form, then we can write the following table by comparing the values of a1/a2, b1/b2, and c1/c2 in all three examples.

Pair of lines a1/a2 b1/b2 c1/c2 Comparison Graphical Representation Algebraic Interpretation
x – 2y = 0 1/3 – 2/4 0/20 (a1/a2) ≠ (b1/b2) Intersecting lines One solution
3x + 4y – 20 = 0
2x + 3y – 9 = 0 2/4 3/6 9/18 (a1/a2) = (b1/b2) = (c1/c2) Coincident lines Infinite solutions
4x + 6y – 18 = 0
x + 2y – 4 = 0 1/2 2/4 4/12 (a1/a2) = (b1/b2) ≠ (c1/c2) Parallel lines No solution
2x + 4y – 12 = 0

From the table above, you can observe that if the lines represented by the equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are,

  1. Intersecting, then (a1/a2) ≠ (b1/b2)
  2. Coincident, then (a1/a2) = (b1/b2) = (c1/c2)
  3. Parallel, then (a1/a2) = (b1/b2) ≠ (c1/c2)

It is important to note that the converse is also true.

Solved Examples for You

Question 1: Check graphically whether the pair of equations ‘x + 3y = 6’ and ‘2x – 3y = 12’ is consistent. If so, solve them graphically.

Answer : Let us draw the graphs of both the equations. The tables for these equations are:

x 0 6
y = (6 – x)/3 2 0

 

x 0 3
y = (2x – 12)/3 – 4 – 2

Now, take a graph paper and plot the following points:

  • A (0, 2)
  • B (6, 0)
  • P (0, – 4)
  • Q (3, – 2)

Next, draw the lines AB and PQ as shown below.

equations in two variables

From the figure above, you can see that the two lines intersect at the point B (6, 0). Therefore, x = 6 and y = 0 is the solution of this pair of equations in two variables. Hence, it is Consistent.

Question 2:  Explain how can you solve an equation with two variables?

Answer: The procedure for solving systems of algebraic equations containing two variables involves beginning by moving the variables to the equation’s different sides. Afterwards, there should be a division of both sides of the equation by one of the variables in order to accomplish the solution for that variable. Afterwards, there should be plugging of that number into the formula.

Question 3: Explain two variable equation?

Answer: A linear system of two equations that consist of two variables refers to any system that a person can write in the form of ax + by = p and cx + dy = q.

Question 4: Explain the way of solving the equations system?

Answer: One can solve equations system with the help of following steps:

  • Solve for one of the existing variables. First, solve for y.
  • Place that equation and solve for x.
  • Substitute x = 4 x = 4 x=4 and finally solve for y.

Question 5: Explain what you understand by a nonlinear equation?

Answer: A system of nonlinear equations refers to a system of multiple equations in multiple variables that have at least one equation that does not happen to be linear.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

2
Leave a Reply

avatar
2 Comment threads
0 Thread replies
1 Followers
 
Most reacted comment
Hottest comment thread
1 Comment authors
Ashish Shukla Recent comment authors
  Subscribe  
newest oldest most voted
Notify of
Ashish Shukla
Guest
Ashish Shukla

In the section “Pair of Linear Equations in Two Variables” where example of Ram eating Mangoes or Apples is given, first of all it’s copy paste from CBSE book because we still see “rides” word from the original example of Akhila taking rides.
Second the equation itself is WRONG. It should be:

x = (1/2)y AND NOT y=(1/2)x as given in the explanation…

Ashish Shukla
Guest
Ashish Shukla

In “Solved Example” one of the equations is x-y=-42 and NOT x-7y=42 as mentioned…

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour
to ISRO

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.