What is a Homogeneous System of Linear Equations?

S7-SA2-0048

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 a set of linear equations where the constant term in every equation is zero. This means all equations look like 'something with x, y, z = 0'. These systems always have at least one solution, which is when all variables are zero.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya, who scored some runs in a cricket match. If 'x' is Rahul's runs and 'y' is Priya's runs, and their combined runs (x + y) equals 0, that's a homogeneous equation. The only way for them to score 0 runs together is if both Rahul scored 0 runs and Priya scored 0 runs. So, x=0, y=0 is a solution.

Worked Example

Step-by-Step

Let's solve the following homogeneous system:
Equation 1: x + 2y = 0
Equation 2: 3x + 6y = 0

Step 1: Notice both equations have 0 on the right side. This confirms it's a homogeneous system.
---Step 2: From Equation 1, we can write x = -2y.
---Step 3: Substitute this value of x into Equation 2: 3(-2y) + 6y = 0.
---Step 4: Simplify the equation: -6y + 6y = 0.
---Step 5: This gives 0 = 0, which is always true. This means there are infinitely many solutions.
---Step 6: We know x = -2y. So, if we choose y = 1, then x = -2. If y = 2, then x = -4. The solution is (x, y) = (-2k, k) where k is any real number.
---Step 7: The trivial solution (0,0) is always present. For k=0, x=0, y=0.

Answer: The system has infinitely many solutions, given by (x, y) = (-2k, k) for any real number k.

Why It Matters

Understanding homogeneous systems is super important in fields like AI/ML and engineering. Engineers use them to design stable structures or analyze electrical circuits. In climate science, these systems can help model how different factors balance out to zero in certain conditions, helping us understand complex environmental processes.

Common Mistakes

MISTAKE: Assuming homogeneous systems only have the trivial solution (all variables are zero). | CORRECTION: While the trivial solution (all variables equal to zero) always exists, homogeneous systems can also have infinitely many non-trivial solutions.

MISTAKE: Forgetting that for a system to be homogeneous, ALL constant terms must be zero. | CORRECTION: If even one equation has a non-zero constant term, it's a non-homogeneous system, and different rules apply.

MISTAKE: Confusing the number of equations with the number of variables when determining solution types. | CORRECTION: The relationship between the number of equations and variables, along with the determinant of the coefficient matrix, determines if there are unique, infinite, or trivial solutions.

Practice Questions

Try It Yourself

QUESTION: Is the system below homogeneous? Give a reason.
x + y = 0
x - y = 0
| ANSWER: Yes, because the constant term in both equations is zero.

QUESTION: Find the trivial solution for the system:
2x + 3y - z = 0
x - y + 4z = 0
| ANSWER: x=0, y=0, z=0

QUESTION: For the system x + y = 0 and 2x + 2y = 0, find a non-trivial solution. | ANSWER: One possible non-trivial solution is x=1, y=-1 (or any multiple of this, like x=2, y=-2).

MCQ

Quick Quiz

Which of the following is true for a homogeneous system of linear equations?

It always has a unique non-zero solution.

It always has the trivial solution (all variables are zero).

It never has any solutions.

The constant terms must all be non-zero.

The Correct Answer Is:

A homogeneous system is defined by having all constant terms equal to zero, which means setting all variables to zero will always satisfy every equation, hence the trivial solution always exists. It can also have non-trivial solutions.

Real World Connection

In the Real World

In designing the suspension system for an electric vehicle (EV), engineers use homogeneous systems to find the 'natural frequencies' or 'modes' where the system is stable when no external forces are applied. This helps ensure a smooth ride even on bumpy Indian roads. Similarly, in financial modeling, if you're trying to balance a budget to exactly zero, homogeneous systems help find the combinations of investments and expenses that achieve this balance.

Key Vocabulary

Key Terms

HOMOGENEOUS: all constant terms are zero | TRIVIAL SOLUTION: the solution where all variables are zero | NON-TRIVIAL SOLUTION: any solution where at least one variable is not zero | LINEAR EQUATION: an equation where variables are raised to the power of 1

What's Next

What to Learn Next

Next, you should explore 'Non-Homogeneous Systems of Linear Equations'. This will teach you about systems where the constant terms are not zero, and how their solutions can be unique, infinite, or even non-existent. It builds on what you've learned here about different types of solutions.