What is a System of Equations?

S3-SA1-0079

Grade Level:

Class 6

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

Definition

What is it?

A system of equations is a set of two or more equations that involve the same unknown variables. We try to find values for these variables that satisfy all the equations at the same time. Think of it like solving multiple puzzles with some common pieces.

Simple Example

Quick Example

Imagine you go to a chai stall. One cup of chai and one samosa cost Rs. 30. Also, two cups of chai and one samosa cost Rs. 50. This is a system of equations! We have two different statements (equations) about the same items (chai and samosa), and we want to find the individual price of one chai and one samosa.

Worked Example

Step-by-Step

Let's use our chai and samosa example.

Step 1: Define variables. Let 'c' be the price of one chai and 's' be the price of one samosa.

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Step 2: Write down the equations from the problem statement.
Equation 1: c + s = 30 (One chai + one samosa = Rs. 30)
Equation 2: 2c + s = 50 (Two chai + one samosa = Rs. 50)

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Step 3: Look for a way to eliminate one variable. Notice that 's' (samosa price) is present in both equations with the same coefficient (1).

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Step 4: Subtract Equation 1 from Equation 2 to eliminate 's'.
(2c + s) - (c + s) = 50 - 30
2c + s - c - s = 20
c = 20

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Step 5: Now we know the price of one chai (c) is Rs. 20. Substitute this value back into either Equation 1 or Equation 2 to find 's'. Let's use Equation 1:
c + s = 30
20 + s = 30

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Step 6: Solve for 's'.
s = 30 - 20
s = 10

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Answer: The price of one chai is Rs. 20, and the price of one samosa is Rs. 10.

Why It Matters

Systems of equations are super important! They help engineers design bridges, economists predict market trends, and even your phone's GPS figure out the best route. Data scientists use them to build intelligent AI models, and computer scientists use them in graphics and game development.

Common Mistakes

MISTAKE: Only finding the value of one variable (e.g., just 'c' in the chai example) and forgetting to find the other. | CORRECTION: Always check the original problem to make sure you have found values for ALL the unknown variables.

MISTAKE: Making calculation errors when adding or subtracting equations, especially with negative numbers. | CORRECTION: Double-check your arithmetic carefully at each step. Use brackets when subtracting entire equations to avoid sign errors.

MISTAKE: Not defining variables clearly at the start. | CORRECTION: Always state what each variable (like 'x' or 'y' or 'c' or 's') represents before you start writing equations. This makes your work clear and prevents confusion.

Practice Questions

Try It Yourself

QUESTION: Two pens and one notebook cost Rs. 40. One pen and one notebook cost Rs. 25. What is the cost of one pen and one notebook? | ANSWER: One pen costs Rs. 15, one notebook costs Rs. 10.

QUESTION: A farmer has some chickens and goats. He counts 10 heads and 32 legs in total. How many chickens and how many goats does he have? (Hint: Chickens have 2 legs, goats have 4 legs). | ANSWER: 4 chickens, 6 goats.

QUESTION: A bus travels a certain distance. If it travels at 50 km/h, it arrives 1 hour late. If it travels at 60 km/h, it arrives 1 hour early. What is the actual distance the bus needs to cover, and what is the correct time it should take? (Let D be distance, T be correct time. Remember: Distance = Speed x Time) | ANSWER: Distance = 600 km, Correct Time = 11 hours.

MCQ

Quick Quiz

Which of these is NOT a system of equations?

x + y = 10 and x - y = 2

2a + b = 7 and a = 3

5m + 2n = 15

p + 2q = 12 and 3p - q = 5

The Correct Answer Is:

A system of equations needs two or more equations with the same variables. Option C only has one equation (5m + 2n = 15), so it is not a system of equations.

Real World Connection

In the Real World

When you book a cab using an app like Ola or Uber, the app uses complex systems of equations to calculate the fare based on distance, time, and traffic. Even ISRO scientists use these systems to plot satellite orbits and ensure they reach their destinations accurately!

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal, usually with an '=' sign. | VARIABLE: A symbol (like x, y, a, b) that represents an unknown value. | SOLUTION: The values of the variables that make all equations in the system true. | ELIMINATION: A method to solve systems by adding or subtracting equations to remove one variable.

What's Next

What to Learn Next

Great job understanding systems of equations! Next, you can explore different methods to solve them, like the 'substitution method' or 'graphical method'. These methods will give you more tools to solve even tougher real-world problems.