What is a Dependent System of Linear Equations?

S6-SA1-0012

Grade Level:

Class 10

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

Definition

What is it?

A dependent system of linear equations is a set of two or more linear equations where at least one equation can be derived from the others. This means they are essentially the same line (or plane, etc.) and thus have infinitely many solutions. In simpler terms, one equation is just a 'copy' or a multiple of another.

Simple Example

Quick Example

Imagine you are buying samosas. If the first shop tells you '2 samosas cost 20 rupees' and the second shop tells you '4 samosas cost 40 rupees', these two statements give you the exact same information about the price per samosa. They are dependent equations because the second statement is just double the first one, not new information.

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 and y in both equations. In Equation 1, the coefficient of x is 1 and y is 2. In Equation 2, the coefficient of x is 2 and y is 4.
---Step 2: Compare the constant terms. In Equation 1, it's 5. In Equation 2, it's 10.
---Step 3: Try to multiply Equation 1 by a constant to get Equation 2. If we multiply (x + 2y = 5) by 2, we get:
2 * (x) + 2 * (2y) = 2 * (5)
2x + 4y = 10
---Step 4: Notice that the new equation (2x + 4y = 10) is exactly the same as Equation 2. This means Equation 2 provides no new information different from Equation 1.
---Step 5: Since one equation can be obtained by multiplying the other by a constant, the system has infinitely many solutions and is therefore a dependent system.
Answer: The system is dependent.

Why It Matters

Understanding dependent systems is crucial in fields like AI/ML to avoid redundant data, in engineering to design efficient systems, and in physics to model situations accurately. Scientists and engineers use this to ensure their models are not overcomplicated with unnecessary information, leading to better predictions and designs.

Common Mistakes

MISTAKE: Confusing dependent systems with inconsistent systems. | CORRECTION: Dependent systems have infinitely many solutions (same line), while inconsistent systems have no solutions (parallel lines).

MISTAKE: Only checking one variable's coefficients. | CORRECTION: You must check if ALL coefficients (for x, y, and the constant term) are proportional between the equations.

MISTAKE: Assuming a system is dependent just because it has the same variables. | CORRECTION: A system is dependent only if one equation is a scalar multiple of the other, meaning they represent the exact same relationship.

Practice Questions

Try It Yourself

QUESTION: Is the system 3x + y = 7 and 6x + 2y = 14 dependent? | ANSWER: Yes

QUESTION: Determine if the system 2x - 3y = 6 and 4x - 6y = 10 is dependent. | ANSWER: No, it is inconsistent (parallel lines).

QUESTION: For what value of 'k' will the system x + 3y = 8 and 2x + ky = 16 be dependent? | ANSWER: k = 6

MCQ

Quick Quiz

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

The lines intersect at exactly one point.

The lines are parallel and never intersect.

The lines are identical, overlapping completely.

The equations have different variables.

The Correct Answer Is:

A dependent system means the equations represent the same line, resulting in infinitely many solutions where the lines overlap. Options A and B describe unique solutions and no solutions, respectively.

Real World Connection

In the Real World

Imagine you're tracking the fuel efficiency of a new car. If one sensor reports 'the car uses 1 liter of petrol to travel 20 km' and another sensor reports 'the car uses 2 liters of petrol to travel 40 km', these are dependent pieces of information. In ISRO, engineers need to identify such redundancies in sensor data to simplify complex calculations and ensure accurate spacecraft navigation without processing unnecessary, repeated information.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation that forms a straight line when graphed, like y = mx + c | COEFFICIENT: The number multiplied by a variable in an equation, e.g., '3' in 3x | CONSTANT TERM: A number in an equation that does not change, e.g., '5' in x + 2y = 5 | INFINITELY MANY SOLUTIONS: When two lines completely overlap, every point on the line is a solution.

What's Next

What to Learn Next

Now that you understand dependent systems, explore 'Inconsistent Systems of Linear Equations'. This will help you distinguish between systems that have no solutions versus those with infinitely many, completing your understanding of how lines can relate to each other.