What is a System of Linear Equations?

S6-SA1-0005

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A system of linear equations is a collection of two or more linear equations that involve the same set of variables. When we have such a system, we are looking for values of these variables that satisfy ALL the equations simultaneously.

Simple Example

Quick Example

Imagine you go to a shop and buy 2 samosas and 1 chai, and the total bill is Rs. 40. Later, your friend buys 1 samosa and 2 chais from the same shop, and their bill is Rs. 35. This is a system of linear equations! We want to find the individual price of one samosa and one chai.

Worked Example

Step-by-Step

Let's use our samosa and chai example.
Equation 1: 2s + c = 40 (where 's' is samosa price, 'c' is chai price)
Equation 2: s + 2c = 35

Step 1: From Equation 2, let's express 's' in terms of 'c'.
s = 35 - 2c

---Step 2: Substitute this value of 's' into Equation 1.
2(35 - 2c) + c = 40

---Step 3: Simplify and solve for 'c'.
70 - 4c + c = 40
70 - 3c = 40
-3c = 40 - 70
-3c = -30
c = 10

---Step 4: Now that we have 'c' = 10, substitute it back into the expression for 's' (from Step 1).
s = 35 - 2(10)
s = 35 - 20
s = 15

---Step 5: Check your answers with both original equations.
Equation 1: 2(15) + 10 = 30 + 10 = 40 (Correct!)
Equation 2: 15 + 2(10) = 15 + 20 = 35 (Correct!)

Answer: The price of one samosa is Rs. 15 and one chai is Rs. 10.

Why It Matters

Understanding systems of linear equations is super important for many advanced fields! Engineers use them to design bridges and circuits, while AI/ML scientists use them to train smart systems that can recognise faces or understand speech. Doctors even use them to calculate correct medicine dosages for patients.

Common Mistakes

MISTAKE: Solving only one equation and thinking you have the answer. | CORRECTION: The solution must satisfy ALL equations in the system simultaneously. Always check your answer in every equation.

MISTAKE: Making calculation errors when substituting values or rearranging equations. | CORRECTION: Double-check each step of your algebra, especially when dealing with negative numbers or fractions.

MISTAKE: Forgetting to distribute a number outside a bracket during substitution. | CORRECTION: Remember to multiply the number outside the bracket by *every* term inside the bracket.

Practice Questions

Try It Yourself

QUESTION: Solve the system:
x + y = 7
x - y = 3
| ANSWER: x = 5, y = 2

QUESTION: Find the values of 'a' and 'b' for the following system:
2a + b = 11
3a - 2b = 6
| ANSWER: a = 4, b = 3

QUESTION: A mobile data plan costs Rs. X per GB and Rs. Y for a fixed monthly charge. If 5 GB costs Rs. 250 and 10 GB costs Rs. 400, find X and Y.
| ANSWER: X = 30, Y = 100

MCQ

Quick Quiz

Which of the following is NOT a system of linear equations?

x + y = 5 and 2x - y = 1

a = 3b + 2 and 4a - b = 7

x^2 + y = 10 and x - y = 2

p + 2q = 8 and 3p - q = 5

The Correct Answer Is:

Option C contains x^2, which makes it a non-linear equation. A system of linear equations must only contain equations where variables are raised to the power of 1.

Real World Connection

In the Real World

Imagine a delivery app like Swiggy or Zomato. When they decide the best route for a delivery person to pick up food from multiple restaurants and deliver to multiple customers, they use complex systems of linear equations. This helps them minimise travel time and fuel, making deliveries faster and more efficient for everyone!

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation where the highest power of the variable is 1, and its graph is a straight line. | VARIABLE: A symbol (like x, y, a, b) representing an unknown value. | SOLUTION: The set of values for the variables that makes all equations in the system true. | SIMULTANEOUS EQUATIONS: Another term for a system of linear equations, implying they are solved together.

What's Next

What to Learn Next

Great job understanding systems of linear equations! Next, you can explore different methods to solve these systems, like the 'Elimination Method' or 'Graphical Method'. These methods will give you more tools to tackle even trickier problems and see how these equations behave visually.