What is Solving Systems of Linear Equations?

S3-SA1-0280

Grade Level:

Class 6

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

Definition

What is it?

Solving Systems of Linear Equations means finding the values for unknown variables (like 'x' and 'y') that make two or more linear equations true at the same time. Imagine you have two puzzles, and you need to find the same missing pieces that fit both puzzles perfectly.

Simple Example

Quick Example

Suppose you bought 2 samosas and 1 jalebi for Rs 40. Your friend bought 1 samosa and 2 jalebis for Rs 35. To find the exact price of one samosa and one jalebi, you would solve a system of linear equations.

Worked Example

Step-by-Step

Let's find the cost of 1 pen (P) and 1 notebook (N).

EQUATION 1: 2P + 1N = 50 (You bought 2 pens and 1 notebook for Rs 50)
EQUATION 2: 1P + 1N = 35 (Your friend bought 1 pen and 1 notebook for Rs 35)

Step 1: Subtract Equation 2 from Equation 1.
(2P + 1N) - (1P + 1N) = 50 - 35

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Step 2: Simplify the equation.
2P - 1P + 1N - 1N = 15
1P = 15

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Step 3: So, the cost of 1 pen (P) is Rs 15.

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Step 4: Now, substitute the value of P (Rs 15) into Equation 2.
1P + 1N = 35
15 + 1N = 35

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Step 5: Solve for N.
1N = 35 - 15
1N = 20

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Step 6: So, the cost of 1 notebook (N) is Rs 20.

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Answer: One pen costs Rs 15 and one notebook costs Rs 20.

Why It Matters

This concept is super useful for solving problems in science, engineering, and even computer programming. Engineers use it to design buildings, economists use it to understand market prices, and data scientists use it to make predictions. You might even use it in the future if you work with AI or build cool apps!

Common Mistakes

MISTAKE: Adding or subtracting equations incorrectly, especially with negative signs. | CORRECTION: Always pay close attention to the signs (+ or -) of each term when combining equations. It's like balancing a scale – what you do to one side, do to the other.

MISTAKE: Forgetting to substitute the value of the first variable found back into one of the original equations. | CORRECTION: After finding one variable's value (e.g., 'x'), always put it back into an original equation to find the other variable (e.g., 'y'). Don't stop after finding just one!

MISTAKE: Not checking the final answer by putting both values back into ALL original equations. | CORRECTION: Always verify your solution! Plug both 'x' and 'y' values into every original equation. If they work in all of them, your answer is correct.

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: You bought 3 pencils and 2 erasers for Rs 25. Your friend bought 1 pencil and 2 erasers for Rs 15. What is the cost of one pencil and one eraser? | ANSWER: One pencil costs Rs 5, one eraser costs Rs 5.

QUESTION: Two numbers have a sum of 20. If one number is 4 more than the other, what are the two numbers? (Hint: Let the numbers be 'a' and 'b'). | ANSWER: The numbers are 12 and 8.

MCQ

Quick Quiz

Which of these is a system of linear equations?

x + 2y = 7

x + 2y = 7 and x^2 + y = 5

x + 2y = 7 and 3x - y = 1

x + 2y > 7

The Correct Answer Is:

Option C has two linear equations (equations where variables have power 1) that need to be solved together. Option A is a single equation, Option B has a squared term (not linear), and Option D is an inequality.

Real World Connection

In the Real World

Imagine a delivery app like Swiggy or Zomato. To find the shortest route for a delivery person, considering traffic and multiple stops, they often use systems of equations. Even when your mobile phone's GPS tells you the best way to get to your friend's house, it's solving a complex system of equations based on satellite data!

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation where the highest power of the variable is 1, forming a straight line when graphed. | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables that are solved together. | VARIABLE: A letter (like x, y, a, b) that represents an unknown number. | SOLUTION: The values of the variables that make all equations in the system true.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore different methods to solve systems of linear equations, like the substitution method and the elimination method. Knowing these will give you more tools to tackle even trickier problems!