Problem 32 - Calculus for Biology and Medicine

Question

Determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 4 y-3 z=6 \\ 2 y+z=1 \\ y+z=0 \end{array} $$

Step-by-Step Solution

Answer

The system is overdetermined and has no solution.

Step 1: Identify the Number of Equations and Variables

Count the number of equations and the number of variables in the system. We have three equations:\[ \begin{align*} 4y - 3z &= 6, \ 2y + z &= 1, \ y + z &= 0 \end{align*} \]We have two variables, \(y\) and \(z\). Since there are more equations than variables, this system is overdetermined.

Step 2: Solve the System Using Substitution or Elimination

Begin by solving the simpler equations. From the third equation, solve for \(y\):\[ y + z = 0 \implies y = -z \]Substitute \(y = -z\) into the second equation:\[ 2(-z) + z = 1 \implies -2z + z = 1 \implies -z = 1 \]So, \(z = -1\). Substitute \(z = -1\) back into \(y = -z\):\[ y = -(-1) = 1 \]

Step 3: Verify Solution with the First Equation

Check if \(y = 1\) and \(z = -1\) satisfy the first equation:\[ 4y - 3z = 6 \implies 4(1) - 3(-1) = 4 + 3 = 7 \]The solution does not satisfy the first equation, which indicates inconsistency, confirming the system is overdetermined and has no solution.

Key Concepts

Understanding an Overdetermined System

In linear algebra, an overdetermined system is one where there are more equations than unknown variables. This means that not all the equations in an overdetermined system can be satisfied simultaneously. In the given exercise, we have three equations but only two variables: \( y \) and \( z \). Thus, the system is overdetermined.

An overdetermined system can often lead to inconsistencies. If the equations are not compatible with each other, then there might not be a solution that satisfies all the equations at once. For instance, lines in a plane represented by each equation might not intersect at a common point, making it impossible to find a solution that works for all equations simultaneously.

Understanding Variable Substitution

Variable substitution is a common method used to solve systems of equations. It involves expressing one variable in terms of another and then substituting this expression into the other equations. In this exercise, the equation \( y + z = 0 \) was used to express \( y \) in terms of \( z \).
The substitution process is as follows:

• Identify the simplest equation for substitution.
• Express one variable in terms of the other. Here, \( y = -z \).
• Substitute this expression into the other equations to find solutions for the variables.

This technique reduces the number of variables and equations and can help find a potential solution. If only the simpler equations are satisfied and not the more complex ones, it may indicate an inconsistency in the system.

Exploring a System of Equations

A system of equations is a set of two or more equations involving the same set of variables. Solving a system means finding values for the variables that satisfy all the equations in the system. In our exercise, we have three equations involving \( y \) and \( z \).
When solving a system, one can employ various methods like substitution, elimination, or graphing. Each method has its own strengths and can be more suitable depending on the system's complexity. The goal is to reduce the number of variables simultaneously, making it easier to solve the system. In an overdetermined system, even after solving with such methods, some equations might not match, leading to no solutions at all.

Identifying Inconsistency in Systems

Inconsistency in systems of equations arises when there is no single set of solutions that satisfies all the equations. This can happen in an overdetermined system, where the surplus of equations imposes conflicting requirements.
When attempting to solve the given system, we find that the values \( y = 1 \) and \( z = -1 \) satisfy two of the equations but not the third one. This discrepancy means the system is inconsistent, as it is impossible for these values to satisfy all the equations concurrently.
An inconsistent system often surfaces in practical problems where redundant information leads to conflicts, indicating that assumptions or measurements might need revisiting. Recognizing inconsistency helps in redirecting efforts towards refining the system or adjusting initial conditions.