What is a Simultaneous Equation?

S3-SA1-0327

Grade Level:

Class 6

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

Definition

What is it?

A simultaneous equation is a set of two or more equations that involve the same unknown variables. We solve them together to find the values of these variables that satisfy all equations at the same time.

Simple Example

Quick Example

Imagine you go to a chai stall. You buy 2 cups of chai and 1 samosa for Rs 50. Your friend buys 1 cup of chai and 2 samosas for Rs 40. We can write these as two separate equations to find the price of one chai and one samosa.

Worked Example

Step-by-Step

Let's find the price of one chai (C) and one samosa (S) from the example:
Equation 1: 2C + 1S = 50
Equation 2: 1C + 2S = 40

Step 1: From Equation 1, express S in terms of C: S = 50 - 2C
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Step 2: Substitute this value of S into Equation 2: 1C + 2(50 - 2C) = 40
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Step 3: Simplify the equation: C + 100 - 4C = 40
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Step 4: Combine like terms: 100 - 3C = 40
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Step 5: Subtract 100 from both sides: -3C = 40 - 100
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Step 6: Simplify: -3C = -60
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Step 7: Divide by -3 to find C: C = -60 / -3 = 20
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Step 8: Substitute C = 20 back into S = 50 - 2C: S = 50 - 2(20) = 50 - 40 = 10

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

Why It Matters

Simultaneous equations are super important for solving problems with multiple unknowns, just like finding prices or distances. Engineers use them to design bridges, economists use them to understand markets, and even computer programmers use them in game development. Learning this helps you think like a problem-solver for future careers in AI, Data Science, and more!

Common Mistakes

MISTAKE: Students sometimes try to solve each equation separately without considering the other. | CORRECTION: Remember that the variables (like 'x' and 'y') represent the SAME values in both equations, so they must be solved together.

MISTAKE: Errors in substitution, like forgetting to multiply the entire expression when substituting. | CORRECTION: Use brackets when substituting an expression to ensure all terms are multiplied correctly.

MISTAKE: Making calculation mistakes with positive and negative numbers during simplification. | CORRECTION: Double-check your addition, subtraction, multiplication, and division, especially with negative signs.

Practice Questions

Try It Yourself

QUESTION: If x + y = 10 and x - y = 2, find the values of x and y. | ANSWER: x = 6, y = 4

QUESTION: Two pens and one notebook cost Rs 70. One pen and two notebooks cost Rs 80. What is the cost of one pen and one notebook? | ANSWER: Pen = Rs 20, Notebook = Rs 30

QUESTION: A bus travelled 300 km in 5 hours. Part of the journey was at 50 km/h and the rest at 70 km/h. How much time did the bus travel at each speed? (Hint: Let 'x' be time at 50 km/h and 'y' be time at 70 km/h) | ANSWER: 2.5 hours at 50 km/h, 2.5 hours at 70 km/h

MCQ

Quick Quiz

Which of these is a pair of simultaneous equations?

x + 5 = 12

y = 3x + 1 and x = 5

2a + b = 15 and a - b = 3

p + q + r = 20

The Correct Answer Is:

Option C has two equations with the same two unknown variables (a and b) that need to be solved together. Options A and B are single equations or one equation and a direct value. Option D has three unknown variables in one equation.

Real World Connection

In the Real World

When a delivery app like Swiggy or Zomato calculates the best route for its riders, it often uses complex simultaneous equations. These equations help figure out the shortest path, how many deliveries can be made, and how much fuel is needed, all while considering traffic and delivery times.

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal. | VARIABLE: A symbol (like x, y) representing an unknown value. | SOLUTION: The value(s) of the variable(s) that make an equation true. | SUBSTITUTION: Replacing a variable with an expression or value.

What's Next

What to Learn Next

Great job understanding simultaneous equations! Next, you can explore different methods to solve them, like the elimination method, which can sometimes be even faster. This will prepare you for more complex problems in algebra!