Problem 29 - Calculus for Biology and Medicine

Question

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} x-2 y+z=3 \\ 2 x-3 y+z=8 \end{array} $$

Step-by-Step Solution

Answer

The system is underdetermined and the solution is given by \(x = t + 5\), \(y = t\), \(z = t - 2\), where \(t\) is any real number.

Step 1: Analyze the System Conditions

A linear system is overdetermined if it has more equations than variables, and underdetermined if it has fewer equations than variables. In the given system, we have two equations and three variables: \(x\), \(y\), and \(z\). This means the system is underdetermined, as there are more variables than equations.

Step 2: Solve for One Variable Using Elimination

We can eliminate one variable by subtracting the first equation from the second. This will help us to express one variable in terms of the others.\[ (2x - 3y + z) - (x - 2y + z) = 8 - 3 \]Simplifying this expression, we have:\[ x - y = 5 \]This equation expresses \(x\) in terms of \(y\): \(x = y + 5\).

Step 3: Solve for Another Variable in Terms of the Remaining

Using the expression for \(x\) from Step 2, substitute it back into the first equation:\[ (y + 5) - 2y + z = 3 \]Simplifying this, we have:\[ -y + z + 5 = 3 \]So:\[ z = y - 2 \].

Step 4: State the General Solution

We now have expressions for \(x\) and \(z\) in terms of \(y\):\[x = y + 5 \z = y - 2\]Therefore, the general solution to the system in terms of a parameter \(y = t\) is:\[x = t + 5, \ y = t, \ z = t - 2\]where \(t\) is any real number.

Key Concepts

Overdetermined System

An overdetermined system in linear algebra occurs when there are more equations than unknowns. This often implies that there might not be a solution that satisfies all of the equations. You can imagine this scenario as having multiple constraints on a certain number of variables, which makes it difficult to find a single solution that fulfills every condition.

To check if a system is indeed overdetermined, count the number of equations and compare them to the number of variables. Here are some characteristics of overdetermined systems:

• More equations than variables.
• It typically does not have a solution that satisfies all equations simultaneously.
• Graphically, in two dimensions, lines or curves representing the equations may not intersect at a single point.

Such systems can still be useful, particularly in sciences like statistics, where approximate solutions (like least squares) are sought rather than precise ones. These approximations help make sense of data that do not precisely fit a perfect model.

Underdetermined System

An underdetermined system is quite the opposite of an overdetermined system. Here, you have fewer equations than unknowns. This generally means infinite solutions are possible, allowing for flexibility in finding solutions.

To classify a system as underdetermined, consider these aspects:

• More variables than equations.
• Infinite solutions due to the lack of sufficient constraints.
• Solutions are often expressed in terms of parameters (like variables "set free").

For instance, in the given exercise, with two equations and three variables, the system is underdetermined. This means we can express two of the variables in terms of the third (often a free parameter). This flexibility is frequently harnessed in problems involving design and optimization to provide broader ranges of solutions.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that involves finding values for variables that satisfy given conditions. There are different methods to solve these equations based on the nature of the system.

Here are a few steps typically involved in solving linear systems:

• Substitution Method: Solve one equation for a variable and substitute this expression in the other equations.
• Elimination Method: Add or subtract equations to eliminate variables, simplifying the system step-by-step.
• Matrix Methods: Use matrices and matrix operations to find solutions, typically useful for larger systems.

In our example, we used elimination to find a relationship between two variables, which simplified our solving process. This led us to express the variables in terms of a parameter, showcasing the nature of solutions in underdetermined systems. The focus was on logical manipulation of equations to express relationships and reach a general solution. Using such techniques, even complex systems can be tackled efficiently.