What is a Dependent System of Equations?

S3-SA1-0118

Grade Level:

Class 7

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

Definition

What is it?

A dependent system of equations is a set of two or more linear equations that have infinitely many solutions. This happens when the equations are actually the same line, just written in different ways. Essentially, one equation can be obtained by multiplying the other equation by a constant.

Simple Example

Quick Example

Imagine you buy samosas. Equation 1 says: 'The cost of 2 samosas is Rs 20.' Equation 2 says: 'The cost of 4 samosas is Rs 40.' These two equations are dependent because if you divide Equation 2 by 2, you get Equation 1. They both tell you that one samosa costs Rs 10, so they have infinite common solutions (any number of samosas at Rs 10 each).

Worked Example

Step-by-Step

Let's check if the following system is dependent:
Equation 1: x + 2y = 5
Equation 2: 2x + 4y = 10

Step 1: Look at the coefficients of x in both equations. In Equation 1, it's 1. In Equation 2, it's 2.
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Step 2: Look at the coefficients of y in both equations. In Equation 1, it's 2. In Equation 2, it's 4.
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Step 3: Look at the constant terms. In Equation 1, it's 5. In Equation 2, it's 10.
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Step 4: Compare the ratios of the coefficients and constants. For x: 2/1 = 2. For y: 4/2 = 2. For constants: 10/5 = 2.
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Step 5: Since all ratios are the same (2), it means Equation 2 is simply Equation 1 multiplied by 2. (2 * (x + 2y) = 2 * 5 => 2x + 4y = 10).
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Step 6: Because one equation is a multiple of the other, they represent the same line and have infinitely many solutions.

Answer: Yes, this is a dependent system of equations.

Why It Matters

Understanding dependent systems helps engineers design stable structures and computer scientists optimize algorithms. In fields like data science and AI/ML, recognizing dependent relationships in data can simplify complex problems and make predictions more accurate. It's like finding a shortcut in a maze!

Common Mistakes

MISTAKE: Confusing dependent systems with inconsistent systems (no solutions). | CORRECTION: Dependent systems have ALL coefficients and constants in proportion, leading to infinite solutions. Inconsistent systems have coefficients in proportion but different constants, leading to no solutions.

MISTAKE: Only checking coefficients of x and y, but not the constant term. | CORRECTION: For a system to be dependent, the ratio of all corresponding coefficients AND the constant terms must be the same.

MISTAKE: Thinking 'dependent' means one equation can be solved without the other. | CORRECTION: 'Dependent' means the equations are essentially the same, so solving one effectively solves the 'other' (which is identical). You don't get a unique solution.

Practice Questions

Try It Yourself

QUESTION: Are these equations dependent? Equation A: 3x - y = 6, Equation B: 6x - 2y = 12 | ANSWER: Yes

QUESTION: Is the system 2x + 3y = 7 and 4x + 6y = 15 dependent? Explain why or why not. | ANSWER: No, it is not dependent. While the x and y coefficients are proportional (4/2 = 2, 6/3 = 2), the constant terms are not (15/7 is not 2). This system is inconsistent (no solution).

QUESTION: Find the value of 'k' that makes the following system dependent: x + 5y = 10 and 3x + ky = 30 | ANSWER: k = 15

MCQ

Quick Quiz

Which of the following describes a dependent system of linear equations?

The equations have exactly one unique solution.

The equations have no solution.

The equations represent the same line and have infinitely many solutions.

The equations have different slopes but the same y-intercept.

The Correct Answer Is:

A dependent system means the equations are essentially identical, representing the same line, which results in infinitely many points (solutions) common to both. Options A and B describe unique and no solutions, respectively.

Real World Connection

In the Real World

Imagine a mobile network provider. If they offer a plan where '1GB costs Rs 10' and another plan that says '2GB costs Rs 20', these are dependent systems. They represent the same underlying rate. Companies use this to simplify pricing or detect redundant data in large databases, like those used by UPI payment systems or for managing cricket match statistics.

Key Vocabulary

Key Terms

Linear Equation: An equation where the highest power of the variable is 1, and its graph is a straight line. | System of Equations: A set of two or more equations with the same variables. | Solution: A set of values for the variables that makes all equations in the system true. | Infinitely Many Solutions: An unlimited number of solutions.

What's Next

What to Learn Next

Great job understanding dependent systems! Next, you should explore 'Inconsistent Systems of Equations'. This will help you understand the three possible outcomes when solving linear equations and how they differ from each other. Keep up the good work!