What is a System of Non-Homogeneous Linear Equations?

S6-SA1-0382

Grade Level:

Class 10

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

Definition

What is it?

A system of non-homogeneous linear equations is a set of linear equations where at least one equation has a non-zero constant term. This means the equations do not all pass through the origin (0,0,...) if graphed. These systems are very common in real-world problems because constant values often exist.

Simple Example

Quick Example

Imagine you buy 2 samosas and 1 chai for Rs 50. Then your friend buys 3 samosas and 2 chais for Rs 85. Here, the cost of samosas (x) and chai (y) are variables, and Rs 50 and Rs 85 are the non-zero constant terms. This forms a non-homogeneous system: 2x + y = 50 and 3x + 2y = 85.

Worked Example

Step-by-Step

Let's solve the samosa and chai problem:
Equation 1: 2x + y = 50
Equation 2: 3x + 2y = 85

Step 1: From Equation 1, express y in terms of x: y = 50 - 2x
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Step 2: Substitute this value of y into Equation 2: 3x + 2(50 - 2x) = 85
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Step 3: Simplify the equation: 3x + 100 - 4x = 85
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Step 4: Combine like terms: -x + 100 = 85
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Step 5: Isolate x: -x = 85 - 100
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Step 6: Solve for x: -x = -15, so x = 15
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Step 7: Substitute the value of x back into the expression for y (from Step 1): y = 50 - 2(15)
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Step 8: Calculate y: y = 50 - 30, so y = 20
Answer: The cost of one samosa (x) is Rs 15 and the cost of one chai (y) is Rs 20.

Why It Matters

Understanding non-homogeneous systems is crucial for solving real-world puzzles in science and engineering. For example, engineers use them to design stable bridges, and AI/ML specialists use them to train smart systems. Doctors also use these equations to figure out medicine dosages based on patient data.

Common Mistakes

MISTAKE: Assuming all constant terms must be zero for the system to be solvable. | CORRECTION: Non-zero constant terms are exactly what makes a system non-homogeneous and they are often solvable. Don't confuse it with homogeneous systems where all constants are zero.

MISTAKE: Forgetting to substitute the value of the first variable back into *one of the original equations* (or the derived expression) to find the second variable. | CORRECTION: Always substitute the first variable's value into an equation to find the value of the other variable. Double-check your calculation.

MISTAKE: Incorrectly combining positive and negative numbers during simplification. For example, 3x - 4x = 7x. | CORRECTION: Pay close attention to signs. 3x - 4x is -x. Practice basic integer arithmetic to avoid these errors.

Practice Questions

Try It Yourself

QUESTION: Solve the system: x + y = 7 and 2x - y = 8. | ANSWER: x = 5, y = 2

QUESTION: A mobile data plan costs Rs 200 for a base amount of data plus Rs 10 per extra GB. Another plan costs Rs 250 for a base amount plus Rs 5 per extra GB. If 'x' is extra GB and 'y' is total cost, write these as a system of non-homogeneous linear equations. | ANSWER: y = 10x + 200; y = 5x + 250

QUESTION: Two friends, Ajay and Bhumi, are saving money. Ajay starts with Rs 100 and saves Rs 20 per day. Bhumi starts with Rs 50 and saves Rs 30 per day. After how many days will they have the same amount of money? What will that amount be? | ANSWER: After 5 days, they will both have Rs 200.

MCQ

Quick Quiz

Which of the following represents a system of non-homogeneous linear equations?

x + 2y = 0; 3x - y = 0

x^2 + y = 5; x - y = 1

4x + 5y = 10; x - 2y = 3

x + y = z; 2x - y = 5

The Correct Answer Is:

Option C has two linear equations (no powers higher than 1) and both equations have non-zero constant terms (10 and 3). Options A is homogeneous, B has a non-linear term (x^2), and D has three variables making it a different type of system.

Real World Connection

In the Real World

Imagine a food delivery app like Swiggy or Zomato. When they calculate the total cost of your order (food price + delivery fee) or optimize delivery routes considering traffic (time = distance/speed + delay), they are often solving systems of non-homogeneous linear equations. These equations help determine the most efficient path or the final bill.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation where the highest power of the variable is 1, like 2x + 3y = 7. | HOMOGENEOUS SYSTEM: A system where all constant terms are zero (e.g., 2x + y = 0). | CONSTANT TERM: A number in an equation that does not change, like the '50' in 2x + y = 50. | VARIABLE: A symbol, usually a letter, representing an unknown value.

What's Next

What to Learn Next

Great job understanding non-homogeneous systems! Next, you can explore the 'Consistency of a System of Linear Equations'. This will teach you whether a system has a unique solution, no solution, or infinitely many solutions, which builds directly on what you've learned here.