What is the Solution to Homogeneous Systems? | Simple Explanation for Class 12

S7-SA2-0383

What is the Solution to Homogeneous Systems using Row Operations?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

A homogeneous system of linear equations is one where all the constant terms are zero. We find solutions to these systems using row operations, which are special steps to simplify the equations without changing their meaning. The goal is to find values for all variables (like x, y, z) that make all equations true.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya, who always share their snacks equally. If Rahul brings 'x' amount of samosas and Priya brings 'y' amount of jalebis, and they both want to end up with zero extra snacks after sharing, this is like a homogeneous system. You're looking for 'x' and 'y' values that make 'x + y = 0' and maybe another similar equation true. The 'zero' on the right side makes it homogeneous.

Worked Example

Step-by-Step

Let's solve the system: x + 2y = 0 and 3x + 6y = 0 using row operations.

Step 1: Write the augmented matrix. This means putting the coefficients of x, y, and the constant terms (which are all zero here) into a matrix format.
[ 1 2 | 0 ]
[ 3 6 | 0 ]

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Step 2: Make the first element of the first row (the '1') a pivot. It already is. Now, make the element below it in the first column a zero. To do this, perform R2 -> R2 - 3*R1 (Row 2 minus 3 times Row 1).
[ 1 2 | 0 ]
[ 3 - 3*1 6 - 3*2 | 0 - 3*0 ]
[ 1 2 | 0 ]
[ 0 0 | 0 ]

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Step 3: The second row is now all zeros. This means we have one less independent equation than variables. From the first row, we get the equation: 1x + 2y = 0, or x + 2y = 0.

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Step 4: Since we have two variables (x, y) but only one useful equation, we can express one variable in terms of the other. Let y = k, where 'k' is any real number. This 'k' is called a parameter.

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Step 5: Substitute y = k into x + 2y = 0.
x + 2k = 0
x = -2k

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Answer: The solution is x = -2k, y = k, where k is any real number. This means there are infinitely many solutions.

Why It Matters

Understanding homogeneous systems is super important for many fields! In AI/ML, these systems help computers find patterns in data for things like face recognition or recommending songs. Engineers use them to design stable bridges and electric vehicles (EVs), while physicists use them to understand how forces balance. It's like finding the 'sweet spot' where everything is in perfect balance.

Common Mistakes

MISTAKE: Forgetting that the constant column in a homogeneous system is always zeros. | CORRECTION: Always write the augmented matrix with a column of zeros on the right side, like [ A | 0 ].

MISTAKE: Not recognizing when there are infinitely many solutions. Students often stop when they get a row of zeros and think there's no solution. | CORRECTION: A row of all zeros means you have a 'free variable'. Assign a parameter (like 'k' or 't') to that variable and express others in terms of it, leading to infinite solutions.

MISTAKE: Performing row operations incorrectly, like adding or subtracting numbers to elements without multiplying the entire row. | CORRECTION: Remember that when you multiply a row by a scalar (number) or add/subtract rows, you must apply the operation to EVERY element in that row, including the zero in the augmented part.

Practice Questions

Try It Yourself

QUESTION: Solve the system: x - y = 0 and 2x - 2y = 0. | ANSWER: x = k, y = k (infinitely many solutions)

QUESTION: Solve the system: x + y + z = 0, y - z = 0, x + 2y = 0. | ANSWER: x = 0, y = 0, z = 0 (only the trivial solution)

QUESTION: Use row operations to solve: x + y - z = 0, 2x - y + z = 0, 3x + 0y + 0z = 0. | ANSWER: x = 0, y = 0, z = 0

MCQ

Quick Quiz

For a homogeneous system of linear equations, which of the following is always true?

It always has exactly one unique solution.

It always has infinitely many solutions.

It always has at least one solution (the trivial solution).

It never has any solution.

The Correct Answer Is:

A homogeneous system always has the 'trivial' solution (all variables equal to zero, e.g., x=0, y=0, z=0). It might also have infinitely many non-trivial solutions, but it can never have no solution.

Real World Connection

In the Real World

Think about how your phone's GPS works. When your phone tries to figure out your exact location, it uses signals from multiple satellites. These signals create a system of equations. If the system were homogeneous, it would be like finding a 'balance point' where all signals perfectly align to pinpoint your position, often helping in calculating errors or ideal paths for delivery apps like Zomato or Swiggy.

Key Vocabulary

Key Terms

Homogeneous System: A system of linear equations where all constant terms are zero, meaning the right side of every equation is 0. | Row Operations: Elementary operations (swapping rows, multiplying a row by a non-zero number, adding a multiple of one row to another) used to simplify matrices. | Augmented Matrix: A matrix formed by combining the coefficient matrix and the constant column of a system of equations. | Trivial Solution: The solution where all variables are zero (e.g., x=0, y=0, z=0). This is always a solution for homogeneous systems. | Parameter: A variable (like 'k') used to represent an infinite set of solutions when there are free variables.

What's Next

What to Learn Next

Great job understanding homogeneous systems! Next, you should explore 'Non-Homogeneous Systems of Linear Equations'. They are similar but have non-zero constant terms, which changes how we find solutions and introduces the possibility of having no solution at all. This will make your understanding of linear algebra even stronger!