Infinite Solutions: Conditions, Graphical Representation and Sample Questions

Content Curator

Infinite solutions represent a countless number of values for variables in an equation. Equations have solutions which totally depend on the number of variables. There are three types of solutions of the equation which are unique solutions, no solution and infinite solution. Unique solution is the only solution of the equation whereas no solution represents the equation without a solution or incorrect equation. Infinite solutions of the equation are boundless sets as every value of a dependent variable might reflect the change in the value of another dependent variable. If the number of variables is more than the number of equations then the equation will have infinite solutions.

Also Read: Pair of Linear Equations in Two variable

Table of Content

Condition for Infinite Solution
Graphical Representation of Infinite Solutions
Things to Remember
Sample Questions

Key Terms: System, Infinite, Consistent, Equation, Pair, Inconsistent, Variable, Constants, Boundless

Condition for Infinite Solution

[Click Here for Sample Questions]

Equations are classified as consistent and inconsistent on the basis of solution. If the system of equations has a unique or infinite solution then, the equation is called consistent. Moreover, the equation with no solution is considered as inconsistent.

Suppose the system of equation with two variables x and y is:

a₁x+b₁y=c₁ ——- (1)

a₂x+b₂y=c₂——- (2)

where a₁, a₂, b₁, b₂, c₁ and c₂ are the constants.

So, the system of equations is consistent and have infinitely many solutions if and only if the below equality holds good

(a₁/a₂) = (b₁/b₂) = (c₁/c₂)

For example, the system of equations given by

2x + 3y = 5

8x + 12y = 20

Has infinitely many solutions because

(2/8) = (3/12) = (5/20) = ¼

Also Read: Consistent Systems of Linear Equation

Graphical Representation of Infinite Solutions

[Click Here for Sample Questions]

Graphical representation of pair of linear equation for the different types of solutions are:

In a graph, if the lines of the equations intersect with each other at one and only one point, then the pair of linear equations have a unique solution.
In the graph, if both the lines are parallel to each other then the equations are inconsistent and have no solution.
In the case of infinite solutions, the graph shows the line may be coincident.

For example, the system of equations given by

2x + 3y = 5

8x + 12y = 20

Because when the latter equation divided by 4, gives the same equation 2x + 3y = 5. Also, both equations coincide with each other. Hence, the pair of equations have infinite solutions.

Graphical Representation of Infinite Solutions

Graphical Representation of Infinite Solutions

Also Read: Linear Equation in Two variable

Things to Remember

[Click Here for Sample Questions]

There are two types of systems of equations: consistent and inconsistent.
Inconsistent system of equations has no solution.
If the number of variables is more than the number of equations then the system has infinite solutions.
The system of equations is consistent and have infinitely many solutions if and only if the below equality holds good

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

In infinite solution, lines are coincident and have the same y-intercept.
The slope of the equation is same for the pair of equations with infinite solutions.

Also Read: Cross Multiplication Method of solving linear equations

Sample Questions

Question 1: What are infinite solutions? (3 Marks)

Answer: Infinite solution represents the countless number of values for the variables in the equation. The equation holds good for any value assigned to the one or more variables.

Equation of one variable x with infinitely many solutions

5 x+25=5(x+5)

Equation of two variable x and y with infinitely many solutions

5 x+7 y=20

10 x+14 y=40

Question 2: How many solutions does the equation 12r+13+5r=13+17r have? (4 Marks)
(1)Unique solution
(2)No solution
(3)Infinite solutions

Answer: (c)

Explanation: Simplifying the equation gives 13+17r=13+17r i.e., left hand side is equal to right hand side. For any value of r, the equation remains the same and equality holds goods.

Let us assume the value of r=2, then the equation is

12(2)+13+5(2)=13+17(2)

24+13+10=13+34

47=47

Therefore, r=2 is the solution of the equation

Let us assume the value of r=5, then the equation is

12(5)+13+5(5)=13+17(5)

60+13+25=13+85

98=98

Therefore, r=5 is the solution of the equation

Hence, the equation has infinitely many solutions.

Question 3: What is the relationship between the ratios of coefficients of equations and the graphical representation of the pair of equations? (3 Marks)

Answer: For the pair of equations given by:

a₁ x+b₁ y=c₁

a₂ x+b₂ y=c₂

where a1, a₂, b₁, b₂, c₁and c₂ are the constants.

The pair of equations on graph,

the lines will be intersecting each other if

(a₁/a₂)≠(b₁/b₂)

the lines will coincide each other if

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

the lines will be parallel if

(a₁/a₂)=(b₁/b₂)≠(c₁/c₂)

Question 4: On comparing the ratios of coefficients, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
(1)5x – 4y+8=0 and 7x+6y – 9=0
(2)9x+3y+12=0 and 18x+6y+24=0 (4 Marks)

Answer: Comparing the equations with the general form:

a₁ x+b₁ y+c₁=0

a₂ x+b₂ y+c₂=0

5x – 4y+8=0 and 7x+6y – 9=0

Here, a₁=5, b₁=-4 and c1=8

And a₂=7, b₂=6 and c₂=-9

Also,

(5/7)≠(-2/3)

Therefore,

(a₁/a₂)≠(b₁/b₂)

Hence, the pair of equations will be intersecting at one point.

9x+3y+12=0 and 18x+6y+24=0

Here, a₁=9, b₁=3 and c₁=12

And a₂=18, b₂=6 and c₂=24

Also,

(9/18)=(3/6)=(12/24)

Therefore,

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

Hence, the pair of equations will coincide.

Question 5: Find the missing value of the coefficient in the equation such that it has infinite solution:
4x+6+__x=10 x+6 (3 Marks)

Answer: The condition on which the equation will have infinite solution is when the left-hand side is equal to right-hand side. So, for every value of the equation, the equality may hold good.

Here, the equation is

4x+6+__x=10 x+6

Let us consider the value of missing coefficient is a. Then,

4x+6+a x=10 x+6

(4+a) x+6=10 x+6

(4+a) x=10 x

(4+a)=10

a=6

Hence the missing value is 6 for the equation to have infinite solution.

Question 6: How to solve the pair of equations? (2 Marks)

Answer: To simplify the pair of equations, there are several methods used such as graphical method, elimination method, substitution method, cross multiplication method, matrix method and determinants method. The common and simplest way to solve the equation is by the use of substitutional or elimination methods. Moreover, the matrix and determinant method is mostly used for the equation with a large number of variables.

Question 7: Make the equations for condition, 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Also, tell whether the equation has unique solution or infinite solutions? (4 Marks)

Answer: Let us assume cost of one pencil is denoted by x and cost of one pen is denoted by y.

5 pencils and 7 pens together cost Rs 50

5 x+7 y=50

7 pencils and 5 pens together cost Rs 46

7 x+5 y=46

So, the pair of equations are 5 x+7 y=50 and 7 x+5 y=46.

Comparing the equations with the general form:

a₁ x+b₁ y+c₁=0

a₂ x+b₂y+c₂=0

Here, a₁=5, b₁=7 and c₁=50

And a₂=7, b₂=5 and c₂=46

Also,

(5/7)≠(7/5)

Therefore,

(a1/a2)≠(b1/b2)

Hence, the pair of equations will have unique solution.

Question 8: Given the linear equation 2x+3y – 8=0, write another linear equation in two variables such that the pair of equation have infinitely many solutions? (3 Marks)

Answer: The system of equations is consistent and have infinitely many solutions if and only if the below equality holds good

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

For the equation 2x+3y – 8=0 to have another equation such that the pair have infinite solution, we have to make the equation such that the above equality holds.

Here, a₁=2, b₁=3 and c₁=-8

Let the equation 6x+9y – 24=0 is paired with it.

Here, a₂=6, b₂=9 and c₂=-24

Also,

(2/6)=(3/9)=(-8/-24)=(1/3)

Therefore, the pair of equations 2x+3y – 8=0 and 6x+9y – 24=0 have infinite solution.

Question 9: What are the characteristics of a system of equations with infinite solution? (3 Marks)

Answer: System of equations with infinite solution have following characteristics:

Number of variables is more than the number of equations.
Lines are coincident and have the same y-intercept.
The slope of the equation is the same.
One equation is multiple of the other.

Question 10: Give an example of an equation with no solution and unique solution? (2 Marks)

Answer: System of equations with no solution is known as inconsistent and has lines which are parallel to each other. The following system of equations with x and y variables have infinite solution:

5 x+7 y=20 and 10 x+14 y=15

Example of the pair of equation with unique solution is:

x+3 y=10 and 3x+5 y=1

Question 11: Show that the system of linear equations has infinite solution:
2x+5y=10 and
10x+25y=50 (5 Marks)

Answer: Given: the pair of equation 2x+5y=10 and 10x+25y=50

To prove: The given system of linear equation has infinite solution i.e., the system of equations is consistent and have infinitely many solutions if and only if the below equality holds good

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

Proof: Comparing the equations with the general form:

a₁ x+b₁y+c₁=0

a₂ x+b₂ y+c₂=0

equations are 2x+5y=10 and 10x+25y=50

Here, a₁=2, b₁=5 and c₁=10

And a₂=10, b2=25 and c2=50

Now, the ratios are:

(a₁/a₂)=2/10=1/5

(b₁/b₂)=5 /25=1/5

(c₁/c₂)=10/50=1/5

Hence,

(2/10)=(5/25)=(10/50)=(1/5)

That is,

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

Therefore, the given system of equations has infinitely many solutions.

Question 12: Prove that the system of equations has infinitely many solutions and also verify it by the graphical representation of the system.
22x+11y=44 and
14x+7y=28 (5 Marks)

Answer: Given: the pair of equation 22x+11y=44 and 14x+7y=28

To prove: The given system of linear equation has infinite solution i.e., the system of equations is consistent and have infinitely many solutions if and only if the below equality holds good

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

Proof: Comparing the equations with the general form:

a₁ x+b₁ y+c₁=0

a₂ x+b₂ y+c₂=0

equations are 22x+11y=44 and 14x+7y=28

Here, a₁=22, b₁=11 and c₁=44

And a₂=14, b₂=7 and c₂=28

Now, the ratios are:

(a₁/a₂)=22/14=11/7

(b₁/b₂)=11 /7=11/7

(c₁/c₂)=44/28=11/7

Hence,

(22/14)=(11/7)=(44/28)=(11/7)

That is,

(a₁/a₂)=(b₁/b₂)=(c₁/c₂)

Therefore, the given system of equation has infinitely many solutions.

Also, verifying the system of equations by graphical representation i.e., the line should coincide.

Prove that the system of equations has infinitely many solutions and also verify it by the graphical representation of the system

Also Read:

Formula of perimeter shapes	Types of triangles	Linear Functions
Geometry formula	Surface areas and volumes revision notes	Algebra
Angle between two planes	Important questions on circle	Triangles
Angle Formula	Collinear Points	Similarity of Triangles
Transversal	Linear Pair Axiom	Angle Between Two Lines
Conjunction	Trapezoid Formula	Trigonometric Identities

Infinite Solutions: Conditions, Graphical Representation and Sample Questions

Condition for Infinite Solution

Graphical Representation of Infinite Solutions

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

3.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

4.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

5.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

Comments

SUBSCRIBE TO OUR NEWS LETTER

Infinite Solutions: Conditions, Graphical Representation and Sample Questions

Condition for Infinite Solution

Graphical Representation of Infinite Solutions

Things to Remember

Sample Questions

CBSE X Related Questions

1.Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

3.Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

4.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

5.Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.
The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

3.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

4.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

5.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.