What is an Inconsistent System? | Simple Explanation for Class 7

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What is an Inconsistent System of Linear Equations (Graphically)?

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

An inconsistent system of linear equations is a set of two or more linear equations that have no common solution. Graphically, this means the lines representing these equations are parallel and never intersect.

Simple Example

Quick Example

Imagine you have two auto-rickshaws, one driven by Ramu Kaka and another by Sita Didi. Ramu Kaka charges Rs 10 plus Rs 5 per kilometer, while Sita Didi charges Rs 15 plus Rs 5 per kilometer. If we try to find a distance where their total fare is the same, we will find it's impossible because Sita Didi always charges Rs 5 more for any distance.

Worked Example

Step-by-Step

Let's check if the system of equations is inconsistent: Equation 1: y = 2x + 3 | Equation 2: y = 2x - 1

1. **Identify the slopes (m) and y-intercepts (c) for each equation (in y = mx + c form).**
For Equation 1: Slope (m1) = 2, Y-intercept (c1) = 3
For Equation 2: Slope (m2) = 2, Y-intercept (c2) = -1

2. **Compare the slopes.**
Here, m1 = m2 = 2. This means the lines are parallel.

3. **Compare the y-intercepts.**
Here, c1 = 3 and c2 = -1. Since c1 is not equal to c2, the lines are distinct (different lines).

4. **Conclusion:** Since the slopes are equal but the y-intercepts are different, the lines are parallel and will never intersect. Therefore, the system has no solution and is inconsistent.

Answer: The system of equations is inconsistent.

Why It Matters

Understanding inconsistent systems helps engineers design structures that won't clash or data scientists understand when different data models cannot agree. It's crucial in fields like AI/ML to identify when a problem has no viable solution, saving time and resources. This knowledge can even help you predict outcomes in physics or economics.

Common Mistakes

MISTAKE: Thinking parallel lines will eventually meet if extended far enough. | CORRECTION: Parallel lines, by definition, never intersect, no matter how far they are extended.

MISTAKE: Confusing inconsistent systems with consistent systems that have many solutions (coincident lines). | CORRECTION: Inconsistent systems have NO solutions (parallel lines). Consistent systems with many solutions have IDENTICAL lines (coincident lines).

MISTAKE: Only checking the slopes and not the y-intercepts. | CORRECTION: To confirm an inconsistent system, both conditions must be met: slopes must be equal AND y-intercepts must be different.

Practice Questions

Try It Yourself

QUESTION: Are the lines represented by y = 3x + 5 and y = 3x - 2 inconsistent? | ANSWER: Yes, because their slopes are equal (3) but their y-intercepts are different (5 and -2).

QUESTION: What type of system is represented by 2x + y = 7 and 4x + 2y = 10? (Hint: Convert to y = mx + c form first) | ANSWER: It is an inconsistent system. (Equation 1: y = -2x + 7; Equation 2: y = -2x + 5. Slopes are equal, y-intercepts are different).

QUESTION: If two lines are parallel, and one passes through (0, 4) and the other through (0, -1), will the system of equations representing them be inconsistent? Explain. | ANSWER: Yes, the system will be inconsistent. Parallel lines have the same slope. Since they pass through different y-intercepts (4 and -1), they are distinct parallel lines and will never intersect, meaning no common solution.

MCQ

Quick Quiz

Which of the following describes an inconsistent system of linear equations graphically?

Lines that intersect at exactly one point

Lines that are parallel and distinct

Lines that overlap completely

Lines that are perpendicular

The Correct Answer Is:

An inconsistent system has no solution, which means the lines representing the equations never intersect. This happens when lines are parallel and distinct (different y-intercepts).

Real World Connection

In the Real World

Imagine two different mobile network providers, Jio and Airtel, offering data plans. If Jio charges Rs 200 for 1GB and Airtel charges Rs 250 for 1GB, and both have the same per-extra-MB charge, then finding a data usage point where their total cost is exactly the same for different base plans is an inconsistent problem. This helps companies understand market competition and customer choices.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation whose graph is a straight line, usually in the form y = mx + c | SLOPE: The steepness of a line, represented by 'm' in y = mx + c | Y-INTERCEPT: The point where a line crosses the y-axis, represented by 'c' in y = mx + c | PARALLEL LINES: Lines that are always the same distance apart and never meet | INCONSISTENT SYSTEM: A system of equations with no solution

What's Next

What to Learn Next

Now that you understand inconsistent systems, you should explore 'Consistent Systems of Linear Equations (Graphically)'. This will teach you about systems that *do* have solutions, either one unique solution or infinitely many solutions, completing your understanding of all possible outcomes for linear equations.