Problem 32 - Calculus for Biology and Medicine

Question

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 4 y-3 z=6 \\ 2 y+z=1 \\ y+z=0 \end{array} $$

Step-by-Step Solution

Answer

The system is overdetermined and inconsistent; no solution satisfies all equations.

Step 1: Identify the Number of Equations and Variables

First, count the number of equations and variables in the system. We have three equations: \(4y - 3z = 6\), \(2y + z = 1\), and \(y + z = 0\), with two variables, \(y\) and \(z\).

Step 2: Determine if the System is Overdetermined or Underdetermined

A system is overdetermined if there are more equations than variables, and underdetermined if there are fewer. Here, with three equations and two variables, the system is overdetermined.

Step 3: Solve the System Using the Simplest Equations

Start solving by substituting one equation into others. From the equation \(y + z = 0\), we can express \(y\) in terms of \(z\): \(y = -z\).

Step 4: Substitute Back into Other Equations

Substitute \(y = -z\) into the other two equations. For the equation \(2y + z = 1\), substitute to get \(2(-z) + z = 1\). Simplifying, we find \(-z = 1\), so \(z = -1\).

Step 5: Solve for Remaining Variables

Substitute \(z = -1\) back into \(y = -z\) to find \(y = -(-1) = 1\).

Step 6: Verify Solution in All Original Equations

Finally, check if this solution satisfies all given equations. For \(4y - 3z = 6\), substitute \(y = 1\) and \(z = -1\): \(4(1) - 3(-1) = 4 + 3 = 7\), which is not equal to 6. The system is inconsistent because not all equations are satisfied.

Key Concepts

Linear Equations

Linear equations form the fundamental building blocks in algebra and appear consistently in various mathematical contexts. A linear equation is an equation between two variables that graphically forms a straight line when plotted on a coordinate plane. The standard form of a linear equation is typically expressed as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. The equation is considered 'linear' due to the highest exponent of variables being one, resulting in a graph that is a straight line.

In our initial system of equations:

• \(4y - 3z = 6\)
• \(2y + z = 1\)
• \(y + z = 0\)

we observe each equation is a linear equation in terms of the variables \(y\) and \(z\). When solving, our goal is to find values for these variables that satisfy each equation's conditions. Understanding and manipulating linear equations is crucial in solving systems of equations.

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. The main objective in solving a system of equations is to find which values of the variables satisfy all equations in the system simultaneously. Systems can consist solely of linear equations or include non-linear equations as well; however, linear systems are the most common and frequently studied.

There are several methods to solve a system of linear equations:

• Substitution: Solving one equation for one variable and substituting this expression into other equations.
• Elimination: Adding or subtracting equations to eliminate a variable, making it easier to solve the remaining equations.
• Graphical method: Plotting each equation on the same graph to find intersections representing solutions.

In our problem, the system has more equations than variables, indicating it's overdetermined. We solved it initially via substitution, leading to finding variables, but revealing that not all original equations can be satisfied simultaneously.

Solving Inconsistent Systems

An inconsistent system of equations occurs when no solution satisfies all equations simultaneously. This situation commonly arises in overdetermined systems where there are more equations than variables, leading to contradictions. In inconsistent systems, solving the equations results in conflicts, such as contradictory results for the same variable.

In the provided system:

• Solving gave us \(y = 1\) and \(z = -1\).
• Upon checking, we found the original equation \(4y - 3z = 6\) did not hold, confirming the system's inconsistency.

Recognizing an inconsistent system is crucial because it saves time by indicating that we should not continue searching for a common solution when it's evident one doesn't exist. This often prompts analysts to reevaluate the system for potential errors or to adjust the original equations or conditions if real-world data collection is involved.