Problem 33 - Calculus for Biology and Medicine

Question

Determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 2 x-7 y+z=2 \\ x+y-2 z=4 \end{array} $$

Step-by-Step Solution

Answer

The system is underdetermined with solutions: \( x = 4 - \frac{6+5z}{9} + 2z \), \( y = \frac{6+5z}{9} \), \( z = z \).

Step 1: Define the Problem

We need to determine if the given system of equations is overdetermined or underdetermined and then solve it. The system is: \( \begin{align*} 2x - 7y + z &= 2 \ x + y - 2z &= 4 \end{align*} \).

Step 2: Count the Equations and Variables

Count the number of equations, which is 2, and the number of variables \((x, y, z)\), which is 3. If the number of variables is greater than the number of equations, the system is underdetermined.

Step 3: Determine the System Type

Since the system has more variables (3) than equations (2), it is underdetermined. This means there may be infinitely many solutions.

Step 4: Solve the System

Express one variable in terms of others from one of the equations. From equation \( x + y - 2z = 4 \), solve for \( x \):\[ x = 4 - y + 2z \]Substitute \( x = 4 - y + 2z \) into the first equation:\[ 2(4 - y + 2z) - 7y + z = 2 \]

Step 5: Simplify and Solve

Expand and simplify the equation:\[ 8 - 2y + 4z - 7y + z = 2 \]Combine like terms:\[ 8 - 9y + 5z = 2 \]This simplifies to:\[ 9y = 6 + 5z \]\[ y = \frac{6 + 5z}{9} \]

Step 6: Express Solution in Parametric Form

We have \( x = 4 - y + 2z \) and \( y = \frac{6 + 5z}{9} \). Substitute \( y \) for its expression:\[ x = 4 - \frac{6 + 5z}{9} + 2z \]This solution can be written in terms of the parameter \( z \). Thus, the general solution is:\[ x = 4 - \frac{6 + 5z}{9} + 2z, \quad y = \frac{6 + 5z}{9}, \quad z = z \]

Key Concepts

Systems of Equations

When dealing with mathematical problems, you will often encounter a set of equations working together to define relationships between variables. These sets are known as "systems of equations." In simpler terms, a system of equations is a collection of two or more equations that involve the same set of variables. The main goal of solving a system of equations is to find the values of the variables that satisfy all the equations in the system simultaneously. There are different scenarios that can occur when dealing with these systems:

• If there is exactly one solution, the system is consistent and independent.
• If there are infinitely many solutions, the system is consistent and dependent.
• If there is no solution, the system is inconsistent.

Systems of equations can be solved using various methods such as substitution, elimination, and graphical representation, each offering its own approach to finding the solutions that apply to all given equations.

Solutions in Parametric Form

In certain systems of equations, especially those that are underdetermined, you may encounter solutions in what's called "parametric form." This form is unique because it uses parameters—often denoted by letters like \( t \) or \( z \)—to express the solutions in a more general way.When a system has more variables than equations, it implies that there might be infinitely many solutions. Instead of pinpointing an exact set of values, we express the solutions in terms of one or more parameters. These parameters represent free variables that can take on any value, providing a complete picture of potential solutions.For example, if solving a system results in:\[x = 4 - \frac{6 + 5z}{9} + 2z, \quad y = \frac{6 + 5z}{9}, \quad z = z\]This form shows us how variables \( x \) and \( y \) depend on the parameter \( z \). The parameter \( z \) can be any real number, essentially describing a line or plane of solutions rather than a single point. Parametric solutions offer flexibility and show all possible solutions to the system.

Variables and Equations Count

Understanding the balance between the number of variables and the number of equations is crucial when analyzing a system of equations. This balance can inform us about the nature of the solutions we might expect. Consider a system with:

• Equal number of equations and variables: There's usually a unique solution, provided the equations are independent.
• More equations than variables: This often leads to an overdetermined system, where not all equations can hold simultaneously, possibly resulting in no solutions.
• More variables than equations: Known as an underdetermined system, this scenario typically results in infinitely many solutions.

In an underdetermined system, the surplus of variables means that not enough equations are present to force a specific value for each variable. As a result, we end up expressing solutions in parametric form, allowing one or more variables to "float," while others are influenced by these free-floating parameters. Keeping track of the variables and equations helps predict the "shape" or nature of the solution space for the system.