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What is a Non-Homogeneous System of Linear Equations?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
A non-homogeneous system of linear equations is a set of equations where at least one equation has a non-zero constant term on the right side. This means the equations are not all equal to zero. They represent real-world problems where outputs or results are not always zero.
Simple Example
Quick Example
Imagine you go to a chai stall. You buy 2 cups of chai and 1 samosa, and the total bill is Rs. 70. This can be written as 2x + 1y = 70, where x is the price of chai and y is the price of a samosa. Since the total (70) is not zero, this is a non-homogeneous equation.
Worked Example
Step-by-Step
Let's solve this system of non-homogeneous equations: x + y = 5 and 2x - y = 4.
Step 1: Write down the equations.
Equation 1: x + y = 5
Equation 2: 2x - y = 4
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Step 2: Add Equation 1 and Equation 2 to eliminate 'y'.
(x + y) + (2x - y) = 5 + 4
x + y + 2x - y = 9
3x = 9
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Step 3: Solve for 'x'.
3x = 9
x = 9 / 3
x = 3
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Step 4: Substitute the value of 'x' into Equation 1 to find 'y'.
x + y = 5
3 + y = 5
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Step 5: Solve for 'y'.
y = 5 - 3
y = 2
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Answer: The solution is x = 3 and y = 2.
Why It Matters
Understanding these systems helps scientists design rockets, engineers build strong bridges, and even economists predict market trends. Doctors use them to calculate medicine dosages, and app developers use them for logistics like food delivery routes. They are fundamental in AI/ML for solving complex problems.
Common Mistakes
MISTAKE: Assuming all non-homogeneous systems always have a unique solution. | CORRECTION: Non-homogeneous systems can have a unique solution, no solution (inconsistent), or infinitely many solutions, just like homogeneous ones.
MISTAKE: Forgetting the constant term when converting to matrix form. | CORRECTION: Always include the constant terms in the augmented matrix (the rightmost column) when representing a non-homogeneous system.
MISTAKE: Confusing non-homogeneous with homogeneous systems. | CORRECTION: Remember, a homogeneous system has all constant terms equal to zero. A non-homogeneous system has at least one constant term that is NOT zero.
Practice Questions
Try It Yourself
QUESTION: Is the system x + 2y = 7 and 3x - y = 1 a non-homogeneous system? | ANSWER: Yes, because the constant terms (7 and 1) are not zero.
QUESTION: Solve the system: x - y = 3 and x + y = 7. | ANSWER: x = 5, y = 2
QUESTION: For what value of 'k' will the system 2x + y = 5 and 4x + 2y = k have infinitely many solutions? | ANSWER: k = 10 (If k=10, the second equation becomes 2(2x+y) = 10, which is 2 times the first equation, meaning they are dependent and have infinite solutions.)
MCQ
Quick Quiz
Which of the following describes a non-homogeneous system of linear equations?
All constant terms are zero.
At least one constant term is non-zero.
It always has a unique solution.
The number of variables must be equal to the number of equations.
The Correct Answer Is:
B
A non-homogeneous system is defined by having at least one equation with a non-zero constant term. Options A describes a homogeneous system. Options C and D are not always true for non-homogeneous systems.
Real World Connection
In the Real World
When you use a navigation app like Google Maps or Ola/Uber, it solves complex non-homogeneous systems behind the scenes. It takes your current location, destination, traffic conditions (which are non-zero factors), and calculates the best route and estimated time of arrival. This is a practical application of solving many equations simultaneously to find the optimal path.
Key Vocabulary
Key Terms
LINEAR EQUATION: An equation where the highest power of the variables is 1, like x + y = 5. | CONSTANT TERM: The number without any variable in an equation, like 5 in x + y = 5. | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables. | HOMOGENEOUS SYSTEM: A system where all constant terms are zero. | SOLUTION: The values of the variables that satisfy all equations in the system.
What's Next
What to Learn Next
Great job learning about non-homogeneous systems! Next, you can explore methods to solve these systems using matrices, like the Cramer's Rule or Gaussian elimination. These methods are super powerful for solving even bigger systems with many variables, which is crucial for advanced topics.
