What is a System of Homogeneous Linear Equations?

S6-SA1-0381

Grade Level:

Class 10

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

Definition

What is it?

A system of homogeneous linear equations is a set of linear equations where the constant term in every equation is zero. This means all equations look like 'ax + by + cz = 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 bought some laddoos and jalebis. If the cost equation for Rahul is 2x + 3y = 0 and for Priya is 4x + 6y = 0, where 'x' is the price of one laddoo and 'y' is the price of one jalebi, this is a system of homogeneous equations. The '0' means they don't have any leftover change or debt related to their purchase.

Worked Example

Step-by-Step

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

Step 1: Notice that Equation 2 (3x + 6y = 0) is just 3 times Equation 1 (3 * (x + 2y) = 3 * 0, which is 3x + 6y = 0). This means the equations are dependent.
---Step 2: From Equation 1, x + 2y = 0, we can express x in terms of y: x = -2y.
---Step 3: Since the equations are dependent, any value of y will give a corresponding x that satisfies both. Let's pick a simple value for y, say y = 1.
---Step 4: Substitute y = 1 into x = -2y. So, x = -2 * 1 = -2.
---Step 5: Check this solution in both equations:
For Equation 1: (-2) + 2(1) = -2 + 2 = 0 (Correct)
For Equation 2: 3(-2) + 6(1) = -6 + 6 = 0 (Correct)
---Step 6: The solution (x, y) = (-2, 1) is one of the infinite solutions. The trivial solution is always (0, 0).
Answer: The system has infinite solutions, including (0, 0) and (-2, 1).

Why It Matters

Understanding homogeneous systems is crucial in fields like AI/ML for tasks such as image recognition and data analysis, where finding patterns that sum to zero is key. In engineering, it helps design stable structures and control systems, ensuring forces balance out. These concepts are used by scientists at ISRO to design rockets and by doctors developing new medicines.

Common Mistakes

MISTAKE: Assuming homogeneous systems only have the (0,0) solution. | CORRECTION: Always remember that while (0,0) is *always* a solution (called the trivial solution), there can be infinite non-trivial solutions if the equations are dependent.

MISTAKE: Forgetting that 'homogeneous' means the constant term is zero. | CORRECTION: If any equation has a non-zero constant term (like 'ax + by = 5'), it is a non-homogeneous system.

MISTAKE: Trying to divide by a variable without considering if it could be zero. | CORRECTION: When simplifying or substituting, be careful with division. It's often safer to use substitution or elimination methods.

Practice Questions

Try It Yourself

QUESTION: Is the system x + y = 0 and 2x - y = 0 a homogeneous system? | ANSWER: Yes, because the constant term in both equations is zero.

QUESTION: Find one non-trivial solution for the system: x - 3y = 0 and 2x - 6y = 0. | ANSWER: Let y = 1. Then x = 3y = 3(1) = 3. So, (3, 1) is a non-trivial solution.

QUESTION: For the system: x + y + z = 0, 2x - y + z = 0, and 3x + 2z = 0, is (0,0,0) a solution? | ANSWER: Yes, (0,0,0) is always a solution for any homogeneous system.

MCQ

Quick Quiz

Which of the following is a characteristic of a system of homogeneous linear equations?

All equations have a non-zero constant term.

It always has exactly one unique solution.

The trivial solution (all variables equal to zero) is always a solution.

It can never have more than two variables.

The Correct Answer Is:

A homogeneous system always has the constant term as zero in every equation, meaning that setting all variables to zero will satisfy all equations. Options A, B, and D are incorrect.

Real World Connection

In the Real World

Imagine a company like Ola or Uber optimizing routes for auto-rickshaws. They use complex mathematical models where finding 'balance points' or 'equilibrium' (which often involve homogeneous systems) helps ensure efficient resource allocation and minimal waiting times for customers across different areas in a city like Mumbai or Delhi.

Key Vocabulary

Key Terms

HOMOGENEOUS: Meaning 'of the same kind' or 'uniform'. In equations, it means all constant terms are zero. | LINEAR EQUATION: An equation where the highest power of any variable is 1. | TRIVIAL SOLUTION: The solution where all variables are equal to zero (e.g., x=0, y=0). It's always a solution for homogeneous systems. | NON-TRIVIAL SOLUTION: Any solution to a homogeneous system where at least one variable is not zero. | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables.

What's Next

What to Learn Next

Next, you can explore 'Systems of Non-Homogeneous Linear Equations'. This will build on what you've learned here by adding non-zero constant terms, making the solutions a bit more varied and interesting to find!