What are Infinitely Many Solutions? | Simple Explanation for Class 10

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What are Infinitely Many Solutions of a System?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

When we have a system of two linear equations, 'infinitely many solutions' means there are countless pairs of (x, y) values that satisfy BOTH equations at the same time. This happens when the two equations actually represent the exact same line, just written in different ways.

Simple Example

Quick Example

Imagine you are buying samosas. One shop owner says, 'Each samosa costs Rs. 10.' Another shop owner says, 'For every 2 samosas, you pay Rs. 20.' Both statements mean the same thing: the price per samosa is Rs. 10. Any number of samosas you buy will satisfy both 'rules' because they are identical. This is like having infinitely many solutions for the price and quantity.

Worked Example

Step-by-Step

Let's check if the system of equations below has infinitely many solutions:
Equation 1: 2x + 3y = 6
Equation 2: 4x + 6y = 12

Step 1: Compare the coefficients of x, y, and the constant terms.
---Step 2: For Equation 1: a1 = 2, b1 = 3, c1 = 6.
---Step 3: For Equation 2: a2 = 4, b2 = 6, c2 = 12.
---Step 4: Calculate the ratios: a1/a2 = 2/4 = 1/2. b1/b2 = 3/6 = 1/2. c1/c2 = 6/12 = 1/2.
---Step 5: Since a1/a2 = b1/b2 = c1/c2 (all are 1/2), the lines are coincident (overlap completely).
---Answer: This system has infinitely many solutions.

Why It Matters

Understanding infinitely many solutions is crucial in fields like Engineering and AI/ML, where systems often need to find optimal outcomes. For example, in Robotics, if a robot has too many ways to achieve a task, engineers need to understand these 'infinite solutions' to program the most efficient path. It's used in designing efficient circuits and even in complex Physics problems.

Common Mistakes

MISTAKE: Confusing infinitely many solutions with 'no solution'. | CORRECTION: Infinitely many solutions means the lines are identical (overlap), while 'no solution' means the lines are parallel and never meet.

MISTAKE: Forgetting to check the ratio of constant terms (c1/c2). | CORRECTION: To confirm infinitely many solutions, all three ratios (a1/a2, b1/b2, and c1/c2) must be equal. If only a1/a2 = b1/b2 but not c1/c2, it's a case of 'no solution'.

MISTAKE: Thinking that any two equations with the same numbers will have infinitely many solutions. | CORRECTION: The coefficients must be proportional. For example, x+y=5 and 2x+2y=10 have infinitely many solutions, but x+y=5 and x+2y=10 do not.

Practice Questions

Try It Yourself

QUESTION: Does the system x + y = 3 and 3x + 3y = 9 have infinitely many solutions? | ANSWER: Yes

QUESTION: For what value of 'k' will the system 2x + 3y = 7 and 6x + 9y = k have infinitely many solutions? | ANSWER: k = 21

QUESTION: Two friends, Ajay and Bala, bought some pens and notebooks. Ajay bought 2 pens and 4 notebooks for Rs. 100. Bala bought 4 pens and 8 notebooks for Rs. 200. Formulate linear equations for this situation and determine if there are infinitely many solutions for the price of a pen and a notebook. | ANSWER: Equations: 2P + 4N = 100 and 4P + 8N = 200. Yes, there are infinitely many solutions because the equations are coincident (4P+8N=200 is just 2 * (2P+4N=100)).

MCQ

Quick Quiz

Which condition indicates that a system of two linear equations (a1x + b1y = c1 and a2x + b2y = c2) has infinitely many solutions?

a1/a2 = b1/b2 ≠ c1/c2

a1/a2 ≠ b1/b2

a1/a2 = b1/b2 = c1/c2

a1*a2 = b1*b2 = c1*c2

The Correct Answer Is:

Option C correctly states the condition for infinitely many solutions, where all three ratios of coefficients are equal, meaning the lines are identical. Options A and B represent 'no solution' and 'unique solution' respectively.

Real World Connection

In the Real World

Imagine a logistics company like Zepto or Dunzo planning delivery routes. If two different routes for a delivery person end up being mathematically identical (just described differently), it means there are 'infinitely many ways' to describe that single optimal path. Understanding this helps in writing efficient algorithms for pathfinding in apps and even in complex simulations for Space Technology, like ISRO planning satellite trajectories.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | LINEAR EQUATION: An equation that forms a straight line when graphed | COEFFICIENTS: The numbers multiplying the variables in an equation | COINCIDENT LINES: Two lines that lie exactly on top of each other, sharing all points | PROPORTIONAL: Having a constant ratio to another quantity

What's Next

What to Learn Next

Now that you understand infinitely many solutions, explore 'No Solution of a System'. This will help you differentiate between lines that are identical and lines that are parallel and never meet, completing your understanding of all possible outcomes for a system of linear equations.