Problem 34 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The system is overdetermined and the solution is \( x = \frac{1}{4}, y = \frac{1}{4} \).

Step 1: Identify the number of equations and variables

The system has three equations (\(3x + y = 1\), \(x - y = 0\), \(4x = 1\)) and two variables, \(x\) and \(y\). A system is overdetermined if there are more equations than variables.

Step 2: Determine if the system is overdetermined or underdetermined

Since there are three equations and only two variables, the system is overdetermined. This means there might not be a solution that satisfies all equations together.

Step 3: Attempt to solve the system of equations

Let's solve this system step-by-step. Start by using the third equation, \(4x = 1\), to find \(x\):\[ x = \frac{1}{4} \]

Step 4: Substitute \(x\) into other equations

Using \(x = \frac{1}{4}\) from the previous step, substitute it into the first equation, \(3x + y = 1\):\[ 3\left(\frac{1}{4}\right) + y = 1 \]Simplifying, we get:\[ \frac{3}{4} + y = 1 \]Subtract \(\frac{3}{4}\) from both sides:\[ y = \frac{1}{4} \]

Step 5: Verify with the second equation

Use \(x = \frac{1}{4}\) and \(y = \frac{1}{4}\) in the second equation, \(x - y = 0\):\[ \frac{1}{4} - \frac{1}{4} = 0 \]This equation holds true, confirming the solution.

Key Concepts

Overdetermined System

An overdetermined system of equations arises when there are more equations than variables. This situation often occurs in practical applications, where too many constraints are placed on a set of unknowns. In such systems, finding a single solution that satisfies every equation is challenging, and sometimes impossible. When dealing with overdetermined systems:

• Identify the number of equations and compare it to the number of variables.
• Understand that having more equations means more constraints, potentially leading to inconsistencies.
• Recognize that an overdetermined system may have no solution, one solution, or an infinite number of solutions; though typically, no exact solution satisfies all the equations simultaneously.

By analyzing the system mentioned, we observe three equations and two unknowns, indicating an overdetermined nature. This tells us that a careful approach is essential to explore the possibilities of finding a consistent solution. Try using techniques like least squares or matrix approaches when standard methods fall short.

Systems of Equations

A system of equations consists of a set of equations with the same variables. The goal is to find values for the variables that make all the equations true simultaneously. Systems of equations can be of different types:

• Linear: Involves equations of the first order, like the ones we are examining in the given question.
• Non-linear: Includes equations with variables raised to higher powers or involving trigonometric, logarithmic functions, etc.

Linear systems like the one here are usually represented in a canonical form. For example, using matrices can simplify operations and make computations more efficient. A matrix form of linear equations allows for the application of algorithms designed to solve these systems. Understanding systems of equations is crucial for modeling complex scenarios in engineering, physics, and economics.

Solving Equations

Solving equations, especially in a system, requires a strategic approach. Since the system in our exercise is overdetermined, solving it involves testing various tactics. Here are ways to tackle such a system:

• Substitution: Solve one of the equations for one variable and substitute it into the others, like using the equation \( 4x = 1 \) to find \( x = \frac{1}{4} \).
• Elimination: Combine equations to eliminate one of the variables, making it easier to solve for others.
• Graphical Method: Plotting each equation on a graph can give a visual representation of where they intersect, indicating potential solutions.

In this exercise, after determining \( x \), values were substituted back to solve for \( y \), followed by verification using other equations. Solving strategies depend on the type of system and specific problem constraints, requiring flexibility and understanding of various methods.

Variables and Equations

Variables are symbols in equations that represent unknowns we need to solve for, like \( x \) and \( y \) in our problem. Equations define relationships between these variables. Key points to remember include:

• Defining Variables: Clearly define and distinguish between different variables; assign them symbols.
• Interdependencies: Variables in a system of equations are related. Solving one can help find others.
• Constraints: Each equation adds a constraint that must be satisfied, influencing the possible values the variables can have.

Understanding the structure of an equation as it relates to its variables helps in developing methods to find solutions. Each equation can be seen as a piece of a puzzle that, when combined with others, forms a complete picture of the relationships expressed within the system. This awareness is fundamental in progressing through systems of equations and finding satisfactory solutions.