What is the Solution of a Non-Homogeneous System?

S6-SA1-0184

Grade Level:

Class 10

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

Definition

What is it?

The solution of a non-homogeneous system of linear equations tells us the values of variables that satisfy all equations together. Unlike homogeneous systems which always have a 'zero' solution, non-homogeneous systems might have a unique solution, infinitely many solutions, or no solution at all.

Simple Example

Quick Example

Imagine you go to a snack shop (kirana store) and buy 2 packets of biscuits and 3 packets of chips for a total of Rs 70. Another day, your friend buys 4 packets of biscuits and 6 packets of chips for Rs 140. This is a non-homogeneous system because the total cost (Rs 70 and Rs 140) is not zero. Finding the price of one biscuit packet and one chip packet is solving this system.

Worked Example

Step-by-Step

Let's find the solution for the system:
Equation 1: x + y = 5
Equation 2: 2x + 3y = 12

Step 1: From Equation 1, express x in terms of y. So, x = 5 - y.
---Step 2: Substitute this value of x into Equation 2. We get 2(5 - y) + 3y = 12.
---Step 3: Simplify the equation: 10 - 2y + 3y = 12.
---Step 4: Combine like terms: 10 + y = 12.
---Step 5: Isolate y: y = 12 - 10, so y = 2.
---Step 6: Substitute the value of y back into the expression for x (from Step 1): x = 5 - 2.
---Step 7: Calculate x: x = 3.
---Step 8: The solution is x = 3, y = 2. This means there is a unique solution.

Why It Matters

Solving non-homogeneous systems is super important in many fields! Engineers use it to design bridges and buildings, ensuring they are stable. In AI/ML, it helps train algorithms to recognize patterns, like distinguishing between a dog and a cat in pictures. Doctors use it to calculate medicine dosages, making sure patients get the right amount.

Common Mistakes

MISTAKE: Forgetting to check if the solution works for ALL equations. | CORRECTION: Always substitute your found values back into every original equation to verify they hold true.

MISTAKE: Confusing non-homogeneous with homogeneous systems. | CORRECTION: Remember, a non-homogeneous system has at least one equation where the constant term (the number on the right side) is not zero.

MISTAKE: Making calculation errors during substitution or elimination. | CORRECTION: Be very careful with signs (+/-) and basic arithmetic. Double-check each step, especially when multiplying or subtracting.

Practice Questions

Try It Yourself

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

QUESTION: Find the solution for: 2a + b = 10 and 3a - 2b = 1. | ANSWER: a = 3, b = 4

QUESTION: A farmer has chickens and goats. There are 15 heads and 40 legs in total. How many chickens and how many goats does he have? (Hint: Chickens have 1 head and 2 legs, goats have 1 head and 4 legs). | ANSWER: Chickens = 10, Goats = 5

MCQ

Quick Quiz

Which of the following is NOT a possible outcome when solving a non-homogeneous system of linear equations?

A unique solution

Infinitely many solutions

Only the trivial solution (all variables are zero)

No solution

The Correct Answer Is:

A non-homogeneous system can have a unique solution, infinitely many solutions, or no solution. The trivial solution (all variables are zero) is typically associated with homogeneous systems, not non-homogeneous ones unless the constant terms are also zero, making it a homogeneous system.

Real World Connection

In the Real World

Imagine you're an engineer at ISRO (Indian Space Research Organisation) planning a rocket launch. You need to calculate the fuel needed, the trajectory, and the force required to escape Earth's gravity. These calculations involve solving complex non-homogeneous systems to ensure the rocket reaches its destination accurately and safely.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables. | NON-HOMOGENEOUS: An equation or system where at least one equation has a non-zero constant term. | UNIQUE SOLUTION: A single, specific set of values for the variables that satisfies all equations. | INFINITELY MANY SOLUTIONS: An endless number of possible sets of values for the variables that satisfy all equations. | NO SOLUTION: No set of values for the variables exists that can satisfy all equations simultaneously.

What's Next

What to Learn Next

Great job understanding non-homogeneous systems! Next, you can explore 'Matrix Methods for Solving Systems of Equations'. This will show you a powerful and faster way to solve even bigger systems, which is super useful in computer programming and data science.