Problem 30 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The system is underdetermined with a general solution: \((x, y, z) = (y + 2, y, 1 - 2y)\).

Step 1: Understand the Definitions

A system of equations is called overdetermined if there are more equations than variables, and it is underdetermined if there are fewer equations than variables. This system has two equations and three variables \(x\), \(y\), and \(z\).

Step 2: Determine the Type of System

Since there are fewer equations (2) than variables (3), this system is underdetermined.

Step 3: Solve for Two of the Variables

Let's express \(x\) and \(y\) in terms of \(z\). Using Equation 1, \(x-y=2\), we can express \(x\) as \(x = y + 2\).

Step 4: Substitute and Simplify

Substitute \(x = y + 2\) into Equation 2: \((y+2) + y + z = 3\). Simplify this to obtain \(2y + z + 2 = 3\), which further simplifies to \(2y + z = 1\).

Step 5: Express Solutions in Terms of One Variable

From \(2y + z = 1\), express \(z\) in terms of \(y\) as \(z = 1 - 2y\). Therefore, the solution can be expressed as \(x = y + 2\), \(y = y\), \(z = 1 - 2y\).

Step 6: General Solution Interpretation

The general solution is \((x, y, z) = (y + 2, y, 1 - 2y)\), where \(y\) can be any real number. This means there are infinitely many solutions depending on the value of \(y\).

Key Concepts

System of Equations

A system of equations consists of multiple equations that are considered together. Each equation often involves the same set of variables. The goal is to find a solution that satisfies all equations simultaneously. For example, in the original exercise, a system of equations is given by:

• \( x - y = 2 \)
• \( x + y + z = 3 \)

This system involves three variables: \( x \), \( y \), and \( z \). System of equations can vary in complexity, number of variables, and type of solutions. They form the foundation of various applications in linear algebra, where these systems are used to model real-world problems.

Underdetermined Systems

Underdetermined systems have fewer equations than variables. In the given exercise, we encounter an underdetermined system because we have only two equations for three variables. Such systems may not provide a unique solution due to the lack of constraints on all of the variables.When approaching an underdetermined system, we typically express some variables in terms of others. This leads to multiple solutions instead of just one. For the system:

• \( x - y = 2 \)
• \( x + y + z = 3 \)

We can express \( x \) and \( z \) in terms of \( y \):

• \( x = y + 2 \)
• \( z = 1 - 2y \)

In an underdetermined system, variables not explicitly resolved in terms of others are considered parameters that can take any real value, usually leading to multiple solutions.

Infinitely Many Solutions

When an underdetermined system allows variables to be expressed in terms of a free variable (parameter), it often results in infinitely many solutions. This is the case in the original problem, where solutions depend on the variable \( y \), which can take any real number.The solutions are given by:

• \( x = y + 2 \)
• \( y = y \)
• \( z = 1 - 2y \)

Since \( y \) is free to vary across all real numbers, each distinct value of \( y \) generates a different set of solutions for \( x \) and \( z \). Thus, there are infinitely many ordered triples \((x, y, z)\) that satisfy both equations in the system.