What is the Solution of a Homogeneous System?

S6-SA1-0183

Grade Level:

Class 10

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

Definition

What is it?

The solution of a homogeneous system of linear equations refers to the set of values for the variables that make all equations true. A homogeneous system is special because all the constant terms (the numbers on the right side of the equals sign) are zero. This means it always has at least one solution, called the trivial solution.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, and you want to know how many samosas each ate. If the total number of samosas eaten by Rohan (R) and Priya (P) is 0, and also 2 times Rohan's samosas plus 3 times Priya's samosas is 0. So, R + P = 0 and 2R + 3P = 0. The only way this can be true is if Rohan ate 0 samosas and Priya ate 0 samosas. This (R=0, P=0) is the trivial solution.

Worked Example

Step-by-Step

Let's find the solution for the homogeneous system:
x + 2y = 0
3x + 6y = 0
---Step 1: Notice both equations have 0 on the right side, so it's a homogeneous system.
---Step 2: From the first equation, x + 2y = 0, we can write x = -2y.
---Step 3: Substitute this value of x into the second equation: 3(-2y) + 6y = 0.
---Step 4: Simplify the equation: -6y + 6y = 0.
---Step 5: This gives 0 = 0, which means the equations are dependent (one is a multiple of the other). This system has infinitely many solutions.
---Step 6: To express the solution, let y = k (where k is any real number). Then x = -2k.
---Answer: The solutions are of the form (x, y) = (-2k, k), where k is any real number. For example, if k=1, then x=-2, y=1 is a solution.

Why It Matters

Understanding homogeneous systems is crucial in fields like AI/ML for optimizing algorithms and in physics for analyzing forces in equilibrium. Engineers use this to design stable structures, and computer scientists apply it in graphics and data analysis. It's the foundation for solving more complex problems in these exciting careers.

Common Mistakes

MISTAKE: Assuming a homogeneous system always has only the trivial solution (x=0, y=0) | CORRECTION: Remember that a homogeneous system can also have infinitely many non-trivial solutions if the determinant of its coefficient matrix is zero (or if the equations are dependent).

MISTAKE: Forgetting that a homogeneous system *always* has at least one solution | CORRECTION: The trivial solution (all variables equal to zero) is always a valid solution for any homogeneous system. You just need to check if there are other, non-trivial solutions.

MISTAKE: Confusing a homogeneous system with a non-homogeneous system | CORRECTION: A homogeneous system has all constant terms (the numbers on the right side of the equals sign) as zero. A non-homogeneous system has at least one non-zero constant term.

Practice Questions

Try It Yourself

QUESTION: Is x=0, y=0 a solution for the system: 5x - 3y = 0 and x + 2y = 0? | ANSWER: Yes, it is always a solution for a homogeneous system.

QUESTION: Find all solutions for the system: x - y = 0 and 2x - 2y = 0. | ANSWER: The solutions are of the form (k, k), where k is any real number.

QUESTION: Determine if the system has only the trivial solution or infinitely many solutions: 4x + y = 0 and 8x + 2y = 0. | ANSWER: Infinitely many solutions (since the second equation is 2 times the first equation).

MCQ

Quick Quiz

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

It always has exactly one unique solution.

It always has no solution.

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

It always has infinitely many solutions.

The Correct Answer Is:

A homogeneous system always has x=0, y=0, z=0... as a solution, which is called the trivial solution. Therefore, it always has at least one solution. It might also have infinitely many solutions, but never no solution.

Real World Connection

In the Real World

In cricket analytics, if coaches want to find out if specific batting techniques (represented by variables) contribute zero to a team's overall score (a homogeneous system), they use these concepts. Similarly, when ISRO scientists design satellite trajectories, they might encounter homogeneous systems to ensure the satellite remains in a stable orbit, where forces balance out to zero.

Key Vocabulary

Key Terms

HOMOGENEOUS SYSTEM: A system of linear equations where all constant terms are zero. | TRIVIAL SOLUTION: The solution where all variables are equal to zero. | NON-TRIVIAL SOLUTION: Any solution to a homogeneous system where at least one variable is not zero. | LINEAR EQUATION: An equation where the highest power of any variable is 1.

What's Next

What to Learn Next

Great job understanding homogeneous systems! Next, explore 'Non-Homogeneous Systems of Linear Equations'. This will build on what you've learned by introducing systems where the constant terms are not all zero, opening up a whole new world of problem-solving!