What is the Solution of a System of Linear Equations?

S3-SA1-0538

Grade Level:

Class 6

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

Definition

What is it?

The 'solution' of a system of linear equations is a set of values for the variables that makes ALL the equations in the system true at the same time. Think of it as finding the 'magic numbers' that work perfectly for every rule given.

Simple Example

Quick Example

Imagine you buy two items, a pen and a notebook. If you know that 1 pen + 1 notebook costs Rs. 20, and 2 pens + 1 notebook costs Rs. 30, the solution would be finding the individual price of one pen and one notebook that fits both these conditions.

Worked Example

Step-by-Step

Let's find the solution for this system:
Equation 1: x + y = 10
Equation 2: x - y = 2

Step 1: We want to find values for 'x' and 'y' that satisfy both equations.
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Step 2: Add Equation 1 and Equation 2 together. This helps us get rid of 'y':
(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 'x = 6', substitute this value into either Equation 1 or Equation 2. Let's use Equation 1:
6 + y = 10
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Step 6: Solve for 'y':
y = 10 - 6
y = 4
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Step 7: Check if x=6 and y=4 work for both equations:
Equation 1: 6 + 4 = 10 (True!)
Equation 2: 6 - 4 = 2 (True!)
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Answer: The solution to the system is x = 6 and y = 4.

Why It Matters

Understanding solutions to linear equations is super important! Engineers use it to design bridges and buildings, economists use it to predict market trends, and even computer scientists use it in AI and machine learning to make smart predictions. It's a foundational skill for many cool careers!

Common Mistakes

MISTAKE: Finding values that only work for ONE equation. | CORRECTION: The solution must work for ALL equations in the system simultaneously.

MISTAKE: Making calculation errors when adding or subtracting equations. | CORRECTION: Double-check your arithmetic carefully, especially with positive and negative numbers.

MISTAKE: Not checking your answer at the end. | CORRECTION: Always substitute your found values back into ALL original equations to ensure they are true.

Practice Questions

Try It Yourself

QUESTION: Is x=3, y=2 a solution for the system: x + y = 5 and x - y = 1? | ANSWER: Yes

QUESTION: Find the solution for the system: a + b = 7 and a - b = 3. | ANSWER: a=5, b=2

QUESTION: Two friends, Rohan and Priya, have a total of 15 cricket cards. Rohan has 3 more cards than Priya. How many cards does each have? (Hint: Let R be Rohan's cards and P be Priya's cards. Form two equations.) | ANSWER: Rohan has 9 cards, Priya has 6 cards.

MCQ

Quick Quiz

Which of these is a solution for the system: x + y = 8 and x = 5?

x=3, y=5

x=5, y=3

x=8, y=0

x=5, y=5

The Correct Answer Is:

If x=5, then from the first equation, 5 + y = 8, which means y = 3. So, x=5 and y=3 is the correct solution.

Real World Connection

In the Real World

Imagine you're tracking daily sales at a small chai shop. You know the total number of cups sold and the total money earned. If you have two types of chai (e.g., Masala Chai and Ginger Chai) with different prices, you can use systems of linear equations to figure out how many cups of each type were sold! This helps shop owners manage their inventory and plan better.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables. | LINEAR EQUATION: An equation where the highest power of the variable is 1. | VARIABLE: A letter (like x or y) representing an unknown number. | SOLUTION: Values for the variables that make all equations true.

What's Next

What to Learn Next

Great job understanding solutions! Next, you can explore different methods to solve systems of linear equations, like the substitution method or the graphical method. This will give you more tools to tackle trickier problems!