What is a System of Equations with Coincident Lines? | Simple Explanation for Class 7

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What is a System of Equations with Coincident Lines (Infinite Solutions)?

Grade Level:

Class 7

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

Definition

What is it?

A system of equations with coincident lines means you have two or more equations that represent the exact same line when you plot them on a graph. Because they are the same line, every single point on one line is also on the other, leading to infinitely many solutions.

Simple Example

Quick Example

Imagine you have two friends, Priya and Rohan, buying pens. Priya says, '2 pens cost 10 rupees.' Rohan says, '4 pens cost 20 rupees.' If 'x' is the cost of one pen, Priya's equation is 2x = 10, and Rohan's is 4x = 20. Both equations simplify to x = 5, meaning they are the same statement, and any number of pens will always have this consistent price. They are coincident lines.

Worked Example

Step-by-Step

Let's check if the following system has coincident lines: --- Equation 1: 3x + 2y = 6 --- Equation 2: 6x + 4y = 12 --- Step 1: Look at Equation 1: 3x + 2y = 6. --- Step 2: Look at Equation 2: 6x + 4y = 12. --- Step 3: Try to make Equation 2 look like Equation 1. Notice that if you divide every term in Equation 2 by 2, you get (6x/2) + (4y/2) = (12/2). --- Step 4: This simplifies to 3x + 2y = 6. --- Step 5: Now, Equation 2 (after simplification) is exactly the same as Equation 1. --- Step 6: Since both equations represent the identical line, they are coincident lines, and there are infinitely many solutions. --- Answer: The system has infinitely many solutions because the lines are coincident.

Why It Matters

Understanding coincident lines is crucial for building smart AI models and analyzing complex data in fields like data science. Engineers use this concept to ensure systems are well-defined, and economists use it to model market behaviors, helping them make predictions about prices and demand.

Common Mistakes

MISTAKE: Thinking that if the numbers are different, the lines must be different. | CORRECTION: Always simplify or multiply equations to see if they can be made identical. For example, 2x+y=4 and 4x+2y=8 are the same line.

MISTAKE: Confusing coincident lines with parallel lines (no solutions). | CORRECTION: Coincident lines have the same slope AND the same y-intercept. Parallel lines have the same slope but different y-intercepts.

MISTAKE: Believing 'infinite solutions' means any number can be a solution. | CORRECTION: 'Infinite solutions' means any point (x,y) that lies on that specific line is a solution. It doesn't mean (100, 200) will be a solution if it's not on the line.

Practice Questions

Try It Yourself

QUESTION: Are the lines 5x + 10y = 15 and x + 2y = 3 coincident? | ANSWER: Yes

QUESTION: For what value of 'k' will the lines 4x + 6y = 10 and 2x + 3y = k be coincident? | ANSWER: k = 5

QUESTION: The cost of 3 samosas and 2 jalebis is Rs 40. If another shop states that the cost of 6 samosas and 4 jalebis is Rs 80, do these two statements represent coincident lines? Explain why. | ANSWER: Yes, because the second statement (6s + 4j = 80) is simply double the first statement (3s + 2j = 40). Both represent the same price relationship.

MCQ

Quick Quiz

Which of the following pairs of equations represents coincident lines?

x + y = 5 and x + y = 6

2x + 3y = 7 and 4x + 6y = 14

x + 2y = 3 and x - 2y = 3

3x - y = 4 and 6x - 2y = 5

The Correct Answer Is:

Option B (2x + 3y = 7 and 4x + 6y = 14) shows coincident lines because if you multiply the first equation by 2, you get the second equation. This means they are the same line. Other options represent parallel or intersecting lines.

Real World Connection

In the Real World

Imagine a delivery app like Zepto or Swiggy. If two different drivers are given routes that are mathematically identical, meaning they follow the exact same path and stops, then their route equations are coincident. This helps in optimizing deliveries and ensuring efficient service.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | COINCIDENT LINES: Two or more lines that lie exactly on top of each other | INFINITE SOLUTIONS: When every point on a line is a solution, meaning there are countless solutions | SLOPE: The steepness of a line on a graph | Y-INTERCEPT: The point where a line crosses the y-axis

What's Next

What to Learn Next

Now that you understand coincident lines, explore 'Systems of Equations with Parallel Lines (No Solutions)'. This will help you differentiate between lines that never meet and lines that are identical, completing your understanding of different types of solutions for linear equations.