What is a Unique Solution?

S3-SA1-0028

Grade Level:

Class 6

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

Definition

What is it?

A unique solution means there is only one specific answer that makes an equation or problem true. It's like finding the one special key that opens a lock, and no other key works.

Simple Example

Quick Example

Imagine you have a number. If you add 5 to it, you get 10. What is that number? Only one number fits this description: 5. So, 5 is the unique solution to the problem 'x + 5 = 10'.

Worked Example

Step-by-Step

Problem: Your friend gave you some marbles. You already had 7 marbles. Now you have a total of 15 marbles. How many marbles did your friend give you?
---Step 1: Understand the problem. We need to find the number of marbles your friend gave.
---Step 2: Let 'x' be the number of marbles your friend gave you.
---Step 3: Write an equation: Marbles you had + Marbles friend gave = Total marbles. So, 7 + x = 15.
---Step 4: To find 'x', subtract 7 from both sides of the equation: x = 15 - 7.
---Step 5: Calculate the result: x = 8.
---Step 6: Check if it's true: 7 + 8 = 15. Yes, it is!
---Answer: Your friend gave you 8 marbles. This is the unique solution because no other number works.

Why It Matters

Finding unique solutions is super important in many fields! In Computer Science, it helps design efficient algorithms. Engineers use it to build stable structures, and Data Scientists rely on it to make accurate predictions. It's like solving a puzzle where only one correct piece fits, ensuring everything works perfectly.

Common Mistakes

MISTAKE: Thinking there can be multiple correct answers for a simple equation like x + 3 = 7. | CORRECTION: For a linear equation with one variable, there will always be only one unique solution. Focus on isolating the variable correctly.

MISTAKE: Confusing 'no solution' or 'infinite solutions' with a unique solution. | CORRECTION: A unique solution means exactly ONE answer. 'No solution' means no answer works, and 'infinite solutions' means every number works.

MISTAKE: Making calculation errors while solving the equation, leading to a wrong 'unique' answer. | CORRECTION: Always double-check your arithmetic steps. Substitute your answer back into the original equation to verify if it makes the statement true.

Practice Questions

Try It Yourself

QUESTION: If you buy a snack for ₹15 and pay with a ₹20 note, how much change do you get? (Let 'c' be the change) | ANSWER: ₹5

QUESTION: A bus has 30 seats. If 12 seats are empty, how many passengers are on the bus? (Let 'p' be the number of passengers) | ANSWER: 18 passengers

QUESTION: Your mobile data plan gives you 'x' GB per month. You used 5 GB and have 3 GB left. How much data (x) did your plan originally give you? | ANSWER: 8 GB

MCQ

Quick Quiz

Which of the following describes a unique solution?

There are many possible answers that work.

There is only one specific answer that makes the statement true.

There are no answers that work.

The answer is always zero.

The Correct Answer Is:

A unique solution means there is exactly one specific answer. Options A, C, and D describe other types of solution scenarios (infinite, no solution, or a specific value that isn't always true).

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola, it calculates the 'unique' shortest or fastest route from your current location to your destination. It's finding the single best path among countless possibilities, considering traffic and distance, to get your auto-rickshaw or car there efficiently.

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal. | VARIABLE: A symbol (like x or y) used to represent an unknown number. | SOLUTION: The value(s) of the variable(s) that make an equation true. | LINEAR EQUATION: An equation where the highest power of the variable is 1.

What's Next

What to Learn Next

Great job understanding unique solutions! Now, you can explore 'Equations with No Solution' and 'Equations with Infinite Solutions'. This will help you understand that not all problems have just one answer, and some have none at all, which is a super important concept in algebra!