Problem 31 - Calculus for Biology and Medicine

Question

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 2 x-y=3 \\ x-y=4 \\ x-3 y=1 \end{array} $$

Step-by-Step Solution

Answer

The system is overdetermined and has no consistent solution.

Step 1: Identify the System

We are given the following system of equations: \[\begin{align*} 2x - y &= 3 \ x - y &= 4 \ x - 3y &= 1 \end{align*}\] The system consists of three equations involving two variables, \(x\) and \(y\).

Step 2: Determine Type of System

An overdetermined system has more equations than unknowns, while an underdetermined system has fewer equations than unknowns. Here we have three equations and two unknowns, which makes the system overdetermined.

Step 3: Solve the System Using Elimination

Take two equations at a time to eliminate variables. Start with the first pair: \(2x - y = 3\) and \(x - y = 4\). Subtract the second equation from the first: \((2x - y) - (x - y) = 3 - 4\), resulting in \(x = -1\).

Step 4: Substitute Value into Another Equation

Use \(x = -1\) in the equation \(x - 3y = 1\): \(-1 - 3y = 1\). Solve for \(y\) by adding 1 to both sides to get \(-3y = 2\), then divide by \(-3\) to get \(y = -\frac{2}{3}\).

Step 5: Validate Solution with Each Equation

Substitute \(x = -1\) and \(y = -\frac{2}{3}\) into each original equation to verify consistency: For \(2x - y = 3\), \(2(-1) - (-\frac{2}{3}) = -2 + \frac{2}{3} = -\frac{4}{3}\), which does not equal 3. This indicates that due to being overdetermined, this system does not have a consistent solution.

Key Concepts

Overdetermined Systems

In the world of linear algebra, an overdetermined system typically features more equations than unknowns. In simpler terms, when a problem contains too many constraints or conditions, it becomes overdetermined.

For example, consider our set of equations:

• 2x - y = 3
• x - y = 4
• x - 3y = 1

Here, we have three equations but only two variables, which is a classic hallmark of an overdetermined system. In most cases, overdetermined systems do not have a solution that satisfies all the conditions simultaneously. Often, they represent conflicting equations that don't intersect at a common point, making finding a single solution very tricky.

However, there are methods to address overdetermined systems, such as using least squares to find an approximate solution. But it's important to note that not every overdetermined system will always offer a valid, consistent solution.

Elimination Method

The elimination method is one of the key strategies for solving linear equation systems. Imagine trying to 'eliminate' one variable at a time to simplify your system and solve it. It's a systematic way to handle equations that may initially seem a bit tangled.

In the exercise, the elimination method was utilized to simplify the system. To see it in action, consider:

• Start with the first two equations: 2x - y = 3 and x - y = 4.
• Subtract the second equation from the first to isolate one variable: (2x - y) - (x - y) = 3 - 4.
• This subtraction simplifies to x = -1.

Now, continue by substituting x = -1 into another equation to find y. The neat trick of elimination is you transform the system step by step, reducing its complexity. Just remember, if you end up with an impossible statement (like 0 = 1), you've bumped into an inconsistent scenario, which often happens in overdetermined systems.

Inconsistent Systems

An inconsistent system is a collection of equations that lacks a common solution, meaning there's no single point where all conditions meet. These systems, especially when overdetermined, can often propose logical conflicts.

You might notice one red flag of inconsistency when, after simplifying the system as much as possible, you end up with a contradiction—like finding -1 = 1.

In our exercise, after applying the elimination method to solve for variables, substituting -1 for x and -\(\frac{2}{3}\) for y into the original equations led to such a contradiction. Specifically, in the equation 2x - y = 3, substituting gave us -\(\frac{4}{3}\) instead of 3, highlighting the inconsistency.

It's crucial to recognize this, as it implies the system's constraints can't all be true at once. Even in real-world applications, such as engineering or data analysis, detecting inconsistencies early can save a lot of time and effort in problem-solving.