What is a Solution Set?

S3-SA1-0747

Grade Level:

Class 6

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

Definition

What is it?

A solution set is a collection of all possible values that make a mathematical statement (like an equation or inequality) true. Think of it as a list of 'correct answers' for a puzzle.

Simple Example

Quick Example

Imagine you have a box of laddoos. If I say 'Number of laddoos = 5', then the only value that makes this true is 5. So, the solution set is just {5}.

Worked Example

Step-by-Step

Let's find the solution set for the equation: x + 3 = 7

1. Our goal is to find the value of 'x' that makes the equation true.
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2. To get 'x' by itself, we need to remove the '+ 3' from the left side.
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3. We do this by subtracting 3 from both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep it balanced.
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4. So, x + 3 - 3 = 7 - 3
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5. This simplifies to x = 4.
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6. Now, let's check: If x = 4, then 4 + 3 = 7. This is true!
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7. Therefore, the only value of 'x' that satisfies the equation is 4.
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8. The solution set for x + 3 = 7 is {4}.

Why It Matters

Understanding solution sets is like finding the right key for a lock! In data science, engineers use solution sets to find optimal routes for delivery apps like Swiggy. In computer science, it helps coders debug programs by finding values that cause errors. It's crucial for building smart systems!

Common Mistakes

MISTAKE: Not checking if the found value actually works in the original equation. | CORRECTION: Always substitute your answer back into the original equation or inequality to verify it makes the statement true.

MISTAKE: Forgetting to apply operations to both sides of the equation. | CORRECTION: Remember the 'balance scale' rule: whatever you do to one side (add, subtract, multiply, divide), you must do the exact same to the other side.

MISTAKE: Thinking there's always only one answer in a solution set. | CORRECTION: Some equations or inequalities can have multiple solutions, or even no solutions, or infinitely many solutions. The solution set collects ALL of them.

Practice Questions

Try It Yourself

QUESTION: What is the solution set for y - 5 = 10? | ANSWER: {15}

QUESTION: Find the solution set for 2 * z = 14. | ANSWER: {7}

QUESTION: A mobile game costs Rs. 20. If you have Rs. 50, how many games can you buy? Write an inequality and find the solution set for the number of games (g) you can buy. | ANSWER: 20g <= 50. The solution set for whole games is {0, 1, 2}.

MCQ

Quick Quiz

Which of these is the solution set for the equation m + 7 = 12?

{19}

{5}

{7}

{12}

The Correct Answer Is:

To solve m + 7 = 12, subtract 7 from both sides: m = 12 - 7, which means m = 5. So, the solution set is {5}.

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola, it calculates the 'best' route. This involves solving many equations and inequalities to find the optimal path (the solution set) based on traffic, distance, and time. Even predicting cricket match outcomes uses solution sets to find probabilities!

Key Vocabulary

Key Terms

EQUATION: A statement that two expressions are equal, shown by an equals sign (=) | INEQUALITY: A statement that compares two values that are not equal, using symbols like <, >, <=, >= | VARIABLE: A symbol (usually a letter) representing an unknown value | SOLUTION: A value that makes an equation or inequality true

What's Next

What to Learn Next

Great job understanding solution sets! Next, you can explore 'Solving Linear Equations with One Variable' to practice finding these sets for slightly more complex problems. This will help you tackle bigger math challenges with confidence!