What is a Solution?

S3-SA1-0505

Grade Level:

Class 6

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

Definition

What is it?

In maths, a 'solution' is the value or values that make an equation or inequality true. It's like finding the missing piece that fits perfectly to balance a puzzle or a weighing scale.

Simple Example

Quick Example

Imagine you have a cricket score: Rohit scored 50 runs, and Virat scored 'x' runs. If their total runs are 120, then 'x' must be 70 because 50 + 70 = 120. So, 70 is the solution for 'x'.

Worked Example

Step-by-Step

Let's say your mother gave you Rs. 100 for snacks. You bought a samosa for Rs. 20 and a juice for Rs. 'y'. If you have Rs. 50 left, how much did the juice cost?

Step 1: Write down the equation. Total money - cost of samosa - cost of juice = money left.
100 - 20 - y = 50

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Step 2: Simplify the known numbers.
80 - y = 50

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Step 3: To find 'y', we need to isolate it. Subtract 80 from both sides of the equation.
-y = 50 - 80

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Step 4: Calculate the right side.
-y = -30

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Step 5: Multiply both sides by -1 to get 'y' as a positive value.
y = 30

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Answer: The juice cost Rs. 30. So, 30 is the solution.

Why It Matters

Understanding solutions is fundamental! From designing rockets at ISRO to predicting stock market trends in economics, finding the right 'x' is key. Engineers use it to build bridges, and data scientists use it to make sense of huge amounts of information, helping companies like Zomato deliver food efficiently.

Common Mistakes

MISTAKE: Adding a number to one side of the equation and subtracting it from the other. | CORRECTION: Whatever operation you do to one side of the equation (like adding 5), you MUST do the exact same operation to the other side to keep it balanced.

MISTAKE: Forgetting to change the sign of a number when moving it across the equals sign. For example, changing 'x + 3 = 5' to 'x = 5 + 3'. | CORRECTION: When you move a term from one side of the equals sign to the other, its sign must change. So, 'x + 3 = 5' becomes 'x = 5 - 3'.

MISTAKE: Confusing multiplication with addition. For example, solving '2x = 6' by doing '6 - 2'. | CORRECTION: If a number is multiplying the variable (like '2x'), you must divide by that number to isolate the variable. So, '2x = 6' becomes 'x = 6 / 2'.

Practice Questions

Try It Yourself

QUESTION: If you have 15 ladoos and eat 'x' ladoos, and are left with 7, how many ladoos did you eat? | ANSWER: x = 8

QUESTION: Your father's age is 3 times your age. If your father is 36 years old, how old are you? (Let your age be 'y') | ANSWER: y = 12

QUESTION: A train travels 100 km in 2 hours. If it continues at the same speed, how many hours ('t') will it take to travel 250 km? (Hint: First find the speed per hour). | ANSWER: t = 5 hours

MCQ

Quick Quiz

What is the solution for 'z' in the equation: z - 10 = 25?

2.5

-15

The Correct Answer Is:

To solve z - 10 = 25, you need to add 10 to both sides of the equation. So, z = 25 + 10, which means z = 35. Options A, C, and D are incorrect because they don't balance the equation.

Real World Connection

In the Real World

When you use a navigation app like Google Maps or Ola, the app constantly solves equations to find the shortest route or the estimated time of arrival. It takes your current location, destination, traffic data, and road speeds (all variables!) and finds the 'solution' for the best path and time. Even predicting cricket match outcomes involves solving complex equations!

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal, usually with an equals sign (=) | VARIABLE: A symbol (like x, y, z) that represents an unknown value | INEQUALITY: A mathematical statement showing that two expressions are not equal, using symbols like <, >, <=, >= | ISOLATE: To get a variable by itself on one side of an equation

What's Next

What to Learn Next

Great job understanding solutions! Now that you know how to find the value that makes an equation true, you're ready to explore 'Linear Equations in One Variable'. This will teach you more advanced techniques to solve equations with one unknown, building directly on what you've learned here.