What is a Simultaneous Linear Equation?

S3-SA1-0022

Grade Level:

Class 6

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

Definition

What is it?

A simultaneous linear equation means we have two or more linear equations that need to be solved together to find the values of the variables that satisfy ALL the equations at the same time. Think of it as solving a puzzle where you need to find numbers that fit multiple clues at once. Each equation usually has two unknown variables, like 'x' and 'y'.

Simple Example

Quick Example

Imagine you go to a chai stall. You buy 2 cups of chai and 1 samosa, and it costs Rs. 40. Your friend buys 1 cup of chai and 2 samosas, and it costs Rs. 35. Here, the price of one chai (let's say 'c') and one samosa (let's say 's') are unknown. We have two equations: 2c + s = 40 and c + 2s = 35. Finding 'c' and 's' that work for both is solving simultaneous equations.

Worked Example

Step-by-Step

Let's find the numbers 'x' and 'y' for these two equations:
Equation 1: x + y = 10
Equation 2: x - y = 2

Step 1: Write down the equations clearly.
x + y = 10 (Eq. 1)
x - y = 2 (Eq. 2)
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Step 2: Notice that if we add Eq. 1 and Eq. 2, the 'y' terms will cancel out.
(x + y) + (x - y) = 10 + 2
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Step 3: Simplify the addition.
x + y + x - y = 12
2x = 12
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Step 4: Solve for 'x'.
x = 12 / 2
x = 6
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Step 5: Now that we have the value of 'x', substitute it into either of the original equations to find 'y'. Let's use Eq. 1.
6 + y = 10
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Step 6: Solve for 'y'.
y = 10 - 6
y = 4
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Answer: So, the solution is x = 6 and y = 4. You can check by putting these values into both original equations.

Why It Matters

Simultaneous linear equations are super important! They help scientists understand how things move, engineers design bridges, and economists predict market trends. If you're interested in building AI, creating computer games, or even analyzing cricket statistics, understanding these equations is a foundational step.

Common Mistakes

MISTAKE: Students sometimes only find the value of one variable (e.g., just 'x') and forget to find the other ('y'). | CORRECTION: Always remember that for simultaneous equations, you need to find the value for ALL unknown variables mentioned in the problem.

MISTAKE: Making calculation errors when adding or subtracting equations, especially with negative signs. | CORRECTION: Double-check your arithmetic, especially when dealing with subtraction or terms that have minus signs. Work slowly and carefully.

MISTAKE: Substituting the value of a variable back into the wrong equation or making a mistake during substitution. | CORRECTION: After finding one variable, substitute it into one of the ORIGINAL equations and re-check your calculation. It's often good to pick the simpler-looking equation.

Practice Questions

Try It Yourself

QUESTION: Solve for 'a' and 'b':
a + b = 7
a - b = 3 | ANSWER: a = 5, b = 2

QUESTION: Find 'p' and 'q':
2p + q = 11
p - q = 4 | ANSWER: p = 5, q = 1

QUESTION: Two friends, Rohan and Priya, bought fruits. Rohan bought 3 apples and 2 bananas for Rs. 70. Priya bought 1 apple and 2 bananas for Rs. 50. What is the price of one apple and one banana? | ANSWER: Price of one apple = Rs. 20, Price of one banana = Rs. 5

MCQ

Quick Quiz

Which of these pairs of values (x, y) is a solution to the simultaneous equations: x + y = 8 and x - y = 2?

(3, 5)

(5, 3)

(4, 4)

(6, 2)

The Correct Answer Is:

For option B (5, 3): 5 + 3 = 8 (satisfies the first equation) and 5 - 3 = 2 (satisfies the second equation). No other option satisfies both equations at the same time.

Real World Connection

In the Real World

Imagine a delivery app like Swiggy or Zomato. They use simultaneous equations to figure out the best routes for delivery riders, considering factors like traffic, distance, and number of orders. This helps them minimize delivery time and maximize efficiency, ensuring your food reaches you hot and fresh!

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal, usually with an '=' sign. | VARIABLE: A letter (like x, y, a, b) that represents an unknown number. | LINEAR EQUATION: An equation where the highest power of the variable is 1, so its graph is a straight line. | SIMULTANEOUS: Happening or existing at the same time.

What's Next

What to Learn Next

Great job understanding simultaneous linear equations! Next, you can explore different methods to solve them, like the 'substitution method' and 'elimination method' in more detail. These methods will give you powerful tools to solve even more complex problems!