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What is a System of Linear Equations in Two Variables?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A system of linear equations in two variables is when you have two or more straight-line equations, and each equation has two unknown values (like 'x' and 'y'). We try to find specific values for these unknowns that make ALL the equations true at the same time. Think of it as solving two puzzles simultaneously.
Simple Example
Quick Example
Imagine you go to a snack stall. You buy 2 samosas and 1 chai, and pay ₹50. Your friend buys 1 samosa and 2 chai, and pays ₹40. Here, 'price of one samosa' (let's say 's') and 'price of one chai' (let's say 'c') are your two unknowns. You have two equations: 2s + 1c = 50 and 1s + 2c = 40. This is a system of linear equations!
Worked Example
Step-by-Step
Let's find the price of samosas and chai from the example above.
1. **Write the equations:**
Equation 1: 2s + c = 50
Equation 2: s + 2c = 40
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2. **Make one variable the same in both equations (Elimination Method):** Let's try to make the 's' terms the same. Multiply Equation 2 by 2:
(s + 2c = 40) * 2 => 2s + 4c = 80 (This is our new Equation 3)
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3. **Subtract the equations:** Now subtract Equation 1 from Equation 3:
(2s + 4c = 80)
- (2s + c = 50)
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3c = 30
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4. **Solve for 'c':**
3c = 30
c = 30 / 3
c = 10
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5. **Substitute 'c' back into an original equation:** Let's use Equation 1 (2s + c = 50):
2s + 10 = 50
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6. **Solve for 's':**
2s = 50 - 10
2s = 40
s = 40 / 2
s = 20
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**Answer:** So, one samosa costs ₹20 and one chai costs ₹10.
Why It Matters
Understanding systems of equations is super useful! Engineers use them to design bridges and buildings safely, and computer scientists use them to create powerful AI programs that can recognize faces or translate languages. Even economists use them to understand market prices and predict trends, helping us make better decisions about money.
Common Mistakes
MISTAKE: Forgetting to apply a multiplication to ALL terms in an equation when trying to eliminate a variable. | CORRECTION: When you multiply an equation by a number, remember to multiply every single term (on both sides of the equals sign) by that number.
MISTAKE: Making calculation errors when adding or subtracting equations, especially with negative numbers. | CORRECTION: Double-check your arithmetic carefully. It helps to write the subtraction clearly, changing the signs of the terms being subtracted.
MISTAKE: Finding values for x and y, but not checking if they work in BOTH original equations. | CORRECTION: Always substitute your final x and y values back into *both* original equations to ensure they make both equations true. This confirms your solution is correct.
Practice Questions
Try It Yourself
QUESTION: The sum of two numbers is 15, and their difference is 3. Find the two numbers. | ANSWER: The numbers are 9 and 6.
QUESTION: At a cricket match, 3 tickets for adults and 2 tickets for children cost ₹900. If 1 adult ticket and 1 child ticket cost ₹350, what is the price of one adult ticket and one child ticket? | ANSWER: Adult ticket: ₹200, Child ticket: ₹150.
QUESTION: A farmer has chickens and goats. He counts 20 heads and 56 legs in total. Assuming each chicken has 1 head and 2 legs, and each goat has 1 head and 4 legs, how many chickens and goats does the farmer have? | ANSWER: 12 chickens and 8 goats.
MCQ
Quick Quiz
Which of these is a system of linear equations in two variables?
x + y = 5
x^2 + y = 7
2x + 3y = 10 and x - y = 1
x + y + z = 12
The Correct Answer Is:
C
Option C has two equations, and each equation has two variables (x and y) raised to the power of 1 (making them linear). Option A is only one equation, Option B has x^2 which is not linear, and Option D has three variables.
Real World Connection
In the Real World
When you use a food delivery app like Swiggy or Zomato, behind the scenes, complex systems of equations help decide the most efficient route for the delivery person, considering traffic and multiple orders. Even ISRO scientists use these systems to calculate rocket trajectories and satellite orbits, ensuring precise movements in space.
Key Vocabulary
Key Terms
VARIABLE: An unknown value represented by a letter (like x or y) | EQUATION: A mathematical statement showing two expressions are equal | LINEAR: Describes an equation whose graph is a straight line (variables have power 1) | SOLUTION: The values of variables that make an equation or system of equations true
What's Next
What to Learn Next
Great job understanding the basics of systems of linear equations! Next, you can explore different methods to solve these systems, like the 'Substitution Method' or 'Graphical Method'. These methods will give you more tools to solve real-world problems efficiently.
