What is a System of Linear Equations with Infinite Solutions?

S3-SA1-0283

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A system of linear equations with infinite solutions means you have two or more equations that are actually the same line or represent the same relationship. When graphed, these lines lie exactly on top of each other, meaning every point on one line is also on the other. This results in countless possible answers that satisfy all equations.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, buying samosas. Rohan says, 'I bought 2 samosas for 20 rupees.' Priya says, 'I bought 4 samosas for 40 rupees.' If we write these as equations (S = number of samosas, C = cost), Rohan's equation is 2C = 20S, and Priya's is 4C = 40S. Notice Priya's equation is just Rohan's equation multiplied by 2. This means any price and number of samosas that works for Rohan's equation will also work for Priya's, giving infinite possibilities.

Worked Example

Step-by-Step

Let's look at this system of equations:
Equation 1: x + y = 5
Equation 2: 2x + 2y = 10

Step 1: Look at Equation 1: x + y = 5.
---Step 2: Look at Equation 2: 2x + 2y = 10.
---Step 3: Try to simplify Equation 2. Notice that every term in Equation 2 (2x, 2y, and 10) can be divided by 2.
---Step 4: Divide Equation 2 by 2: (2x / 2) + (2y / 2) = (10 / 2).
---Step 5: This simplifies to x + y = 5.
---Step 6: Now compare the simplified Equation 2 (x + y = 5) with Equation 1 (x + y = 5). They are exactly the same!
---Step 7: Since both equations are identical, any pair of numbers (x, y) that adds up to 5 will satisfy both equations. For example, (1, 4), (2, 3), (0, 5), (5, 0), (2.5, 2.5), etc.
Answer: This system has infinite solutions because the equations are essentially the same.

Why It Matters

Understanding infinite solutions helps in fields like AI and Data Science when models have too much flexibility or redundancy. Engineers use this to design systems where multiple input combinations lead to the same output. It's also crucial in economics to model situations with perfectly interchangeable resources.

Common Mistakes

MISTAKE: Thinking 'infinite solutions' means any numbers for x and y will work. | CORRECTION: Infinite solutions means there are many, many pairs of (x,y) that work, but they must all satisfy the specific relationship defined by the equations (e.g., x+y=5). Not just *any* random numbers.

MISTAKE: Confusing infinite solutions with 'no solution'. | CORRECTION: No solution means the lines are parallel and never meet. Infinite solutions means the lines are exactly the same and meet everywhere.

MISTAKE: Not simplifying equations before comparing them. | CORRECTION: Always simplify equations by dividing or multiplying by a common factor to see if they become identical. For example, 2x+2y=10 and x+y=5 are the same.

Practice Questions

Try It Yourself

QUESTION: Does the system x - y = 3 and 3x - 3y = 9 have infinite solutions? | ANSWER: Yes, because if you divide the second equation by 3, you get x - y = 3, which is identical to the first equation.

QUESTION: Find a value for 'k' such that the system x + 2y = 7 and 3x + ky = 21 has infinite solutions. | ANSWER: For infinite solutions, the second equation should be a multiple of the first. If we multiply the first equation by 3, we get 3x + 6y = 21. Comparing this to 3x + ky = 21, we see k must be 6.

QUESTION: Rohan buys 5 pens and 3 notebooks for 100 rupees. His friend, Shreya, says she bought 10 pens and 6 notebooks for 200 rupees. Do these two situations represent a system with infinite solutions for the price of pens and notebooks? Explain. | ANSWER: Yes. Let 'p' be the price of a pen and 'n' be the price of a notebook. Rohan's equation: 5p + 3n = 100. Shreya's equation: 10p + 6n = 200. If you divide Shreya's equation by 2, you get 5p + 3n = 100, which is identical to Rohan's. This means any price combination (p,n) that works for Rohan also works for Shreya, indicating infinite solutions.

MCQ

Quick Quiz

Which of the following systems of equations has infinite solutions?

x + y = 4 and x - y = 2

2x + 3y = 6 and 4x + 6y = 12

x + y = 5 and x + y = 7

x = 3 and y = 5

The Correct Answer Is:

Option B has 2x + 3y = 6 and 4x + 6y = 12. If you divide the second equation by 2, you get 2x + 3y = 6, which is identical to the first. This means they are the same line and have infinite solutions. Other options have one solution (A, D) or no solution (C).

Real World Connection

In the Real World

Imagine you're building a smart home system in India. If you program 'Turn on lights when it's dark' and then also 'Turn on lights when brightness is below 20 lumens' (and 20 lumens means it's dark), these are essentially the same instruction. The system has infinite ways to interpret 'darkness' that trigger the lights, as the two rules are redundant. This is similar to having infinite solutions.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | LINEAR EQUATION: An equation where the highest power of the variable is 1, and its graph is a straight line | VARIABLES: Letters (like x, y) that represent unknown values | SOLUTION: A value or set of values for the variables that makes the equation true | IDENTICAL EQUATIONS: Equations that are essentially the same, even if one is a multiple of the other

What's Next

What to Learn Next

Great job understanding infinite solutions! Next, you should explore systems of linear equations with 'no solutions' and 'one unique solution'. This will complete your understanding of all possible outcomes for linear systems and prepare you for more advanced algebra.