Linear equations are an important part of algebra in mathematics. It has various real-life applications in applied science, other than mathematics. In systems of linear equations, the solutions can be classified into three categories: a single unique solution, no solution, or infinitely many solutions. When the graph line of equations are coincident on each then the given pair of linear equations have infinitely many solutions.

Infinitely many solutions are important for solving complex problems in linear algebra, computer science, and engineering, where systems of equations often model real-world phenomena.

What Are Infinitely Many Solutions?

A system of linear equations has Infinitely many solutions when there are infinite values that satisfy all equations in the system simultaneously. This means that the equations are dependent, and one equation can be derived from another.

The linear equation is represented in standard form:

• a₁x + b₁y + c₁ = 0
• a₂x + b₂y + c₂ = 0

For infinitely many solutions:

If (a₁/a₂) = (b₁/b₂) = (c₁/c₂)

If this condition is satisfied then the given linear equations have infinitely many solutions.

Example: Check whether the pair of equations has infinitely many solutions.

• 3x + 4y = 3
• 6x + 8y = 6

For this pair of equations we can see that:

(a₁/a₂) = (b₁/b₂) = (c₁/c₂)

3/6 = 4/8 = 3/6

So. this pair of equation have infinitely many solutions.

Understanding Infinitely Many Solutions

• Increased problem-solving: When we know how to solve these questions then it will be easier for us to solve complex problems in mathematics.
• Builds Foundation for advanced and complex topics: There are many applications of infinitely many solutions and it is used in many advanced mathematics topics such as linear algebra, differential equations, and calculus.
• Real-World Applications: This concept is used in many real-life life applications such as economics, physics, and engineering.

Systems of Equations in Two Variables having Infinite Solutions

When we take two equations then it must have two different variables present in them. When the planes in the graph of two equations both lie on one another then equations have infinitely many solutions.

Consider the following system:

• 2x + 4y = 6
• 4x + 8y = 12

By simplifying the second equation, we get: [Tex]\frac{4x + 8y}{2} = \frac{12}{2} \Longrightarrow 2x+4y=6 [/Tex]

Since both equations are equivalent, the system has infinitely many solutions.

Systems of Equations in Three Variables having Infinite Solutions

When we take three equations then it must have three different variables present in them. When the planes in the graph of all three equations all lie on one another then equations have infinitely many solutions.

Consider this system:

• x + 2y + 3z = 4
• 2x + 4y + 6z = 8
• 3x + 6y + 9z = 12

By simplifying the second and third equations, we find they are multiples of the first equation, indicating infinitely many solutions.

Solving Systems with Infinitely Many Solutions

There are different methods for Solving systems with infinitely many solutions. Some important methods are mentioned below:

• Graphical Method
• Substitution Method
• Elimination Method

Graphical Method

In the graphical method, we plot each equation on the graph to determine the intersection of lines of different equations. If the lines represented by equations coincide, it indicates that there are infinitely many solutions. For the variable linear system we will draw lines and for the three-variables system, we will draw planes on the graph.

We can interpret the following from the graph:

If intersection at one point: There is only one solution.

If there is no intersection i.e. parallel lines/ planes: No solution exist

If the lines are coincident: There exist infinitely many solutions.

Example: For given Equation,

• y = 2x + 3
• 3x = 6x + 9

Check whether there exist infinitely many solutions.

Solution:

**graph

When we plot both lines on graph then these lines will coincide indicating that there exist infinitely many solutions.

Substitution Method

In this method, we will substitute one equation into another to solve the equation. First, we will choose one of the equations and solve it for one variable in terms of another variable. Now we substitute the above expression in another equation which will reduce the number of variables. We simplify this equation if we get a true identity for any value (for example 0 = 0) then this system of equation has infinitely many solutions.

Example: For given Equation,

• x + 2y = 4
• 2x + 4y = 8

Check whether there exist infinitely many solutions

Solution:

Solve the first equation for x:

x = 4-2y

Substitute this into the second equation:

2(4 – 2y) + 4y = 8

8 – 4y + 4y = 8

8 = 8

Which is a true identity indicating that this system of equations has infinitely many solutions.

Elimination Method

In the elimination method, we will add or subtract the equation to eliminate one of the variables, and simplify until it is solved or easy to solve. This method is used when the coefficients of one variable are the same or multiples of each other. First, we will multiply one or both equations by a particular number to obtain coefficients for one variable that are opposites or equal. Now we add or subtract the equation and eliminate one variable. We simplify this equation if we get a true identity for any value (For example 0 = 0) then this system of equation has infinitely many solutions.

Example: For given Equation,

• 3x + 6y = 12
• 6x + 12y = 24

Check whether there exist infinitely many solutions

Solution:

By observation, we can clearly see that the second equation is just twice the first.

Subtract the first equation from the second:

6x + 12y – (3x + 6y) = 24 – 12

3x + 6y = 12

This simplification shows that the two equations are identical, indicating infinitely many solutions.

Examples on Infinitely Solutions

Example 1: For the given system of equations check whether it has infinitely many solutions:

• 2x + 3y = 6
• 4x + 6y = 12

Solution:

First, write the equations in standard form:

2x + 3y = 6

4x + 6y = 12

Second equation is simply obtained from the first equation when multiplied by 2.

Therefore, they represent the same line and have infinitely many solutions.

Any point (x , y) that satisfies the first equation will also satisfy the second equation.

For example, (0 , 2) (3,0) are both solutions of given system of equations.

Example 2: Find the value of k for which the following system has infinitely many solutions:

• x – 2y = 3
• 2x – 4y = k

Solution:

For the system to have infinitely many solutions, the second equation must be a multiple of the first.

Therefore, if we multiply the first equation by 2, the the equation we get: 2(x – 2y) = 2⋅3

2x – 4y = 6

Thus, the value of k for which given system of equations has infinitely many solutions k = 6.

Example 3: Determine if the system of equations has infinitely many solutions:

• x + y+ z = 1
• 2x + 2y + 2z = 2
• 3x +3y +3z = 3

Solution:

Notice that for the second and third equations are multiples of the first equation. Specifically:

2(x + y + z) = 2

3(x + y + z) = 3

Hence, the three equations are equivalent, indicating infinitely many solutions.

Any set (x, y, z) that satisfies x + y + z = 1 is a solution. For example, (1, 0, 0) , (0, 1, 0), and (0, 0, 1) are solutions.

Example 4: Determine if the following system has infinitely many solutions:

• 3x – 2y = 6
• 6x – 4y = 12

Solution:

Notice that the second equation is a multiple of the first:

6x – 4y = 2(3x – 2y) = 2 × 6 = 12

Thus, the system represents the same line and has infinitely many solutions.

Any (x, y) 3x – 2y = 6 is a solution.

Example 5: Find the solutions to the following system with infinitely many solutions:

• x – y + 2z = 3
• 2x – 2y + 4z = 6
• 3x – 3y + 6z = 9

Solution:

Notice the second and third equations are multiples of the first:

2(x – y + 2z) = 2 × 3 = 6

3(x -y + 2z) = 3 × 3 = 9

Thus, any set (x, y, z) that satisfies x – y + 2z = 3 is a solution, such as (3, 0, 0) or (1, -1, 2).

Practice Questions on Infinitely Solutions

Ques 1: Determine if the system has infinitely many solutions, or not?

• 3x + 6y = 12
• 6x + 12y = 24

Ques 2: Find the value of k that makes the system have infinitely many solutions.

• 2x + 4y = 8
• kx + 2y = 4

Ques 3: Determine if the system has infinitely many solutions, no solution, or a unique solution

• x + 2y + 3z = 6
• 2x + 4y + 6z = 12
• 3x + 6y + 9z =1 8

Ques 4: Convert the following system to an augmented matrix and determine if it has infinitely many solutions:

• x – y +z = 2
• 2x – 2y + 2z = 4
• 3x – 3y + 3z = 6

Does this system have infinitely many solutions?

Ques 5: Given the system:

• 4x + 5y = 20
• 8x + 10y = 40

Explain why this system has infinitely many solutions and describe the geometric interpretation.

Conclusion

In this article we understood about system of equations having infinitely many solutions. This information is useful in fields like engineering, physics, computer science, and economics where solving systems of equations is a frequent activity, in addition to being important to pure mathematics.

We have discussed some practice questions to understand the concept of infinite solutions for given system fo equations. This gives us ability to handle various complex case because infinitely many solutions is an invaluableand important skill in the mathematical toolkit.

Also Read:

• System of Linear Equations
• Number of Solutions to a System of Equations Algebraically

FAQs on Infinitely solutions

How we can identify infinitely many solutions in a given set of linear Equations?

When the two given linear equations are identical then that set of equations has infinitely many solutions. We simplify the equation in a simpler form.

Name some methods which can be used to solve linear system which has infinitely many solutions.

Some methods which can be used to solve systems with infinitely many solutions are – graphical method, substitution method, and elimination method.

What is the importance of infinitely many solutions in real-world applications?

There are different real-life applications of Infinitely many solutions such as they help in modeling complex phenomena in physics, economics, engineering, and other fields.