Problem 118 - Calculus 3

Question

True or False? Justify your answer with a proof or a counterexample. The following system of algebraic equations has a unique solution: \(6 z_{1}+3 z_{2}=8\) \(4 z_{1}+2 z_{2}=4\)

Step-by-Step Solution

Answer

False. The system has infinitely many solutions, not a unique solution.

Step 1: Write the System of Equations

First, we have the system of equations given by: 1. \(6z_1 + 3z_2 = 8\)2. \(4z_1 + 2z_2 = 4\).

Step 2: Check for Proportionality

Let's compare the coefficients of \(z_1\) and \(z_2\) in both equations. For the first equation, the ratio of coefficients is \(\frac{6}{4} = \frac{3}{2}\) for \(z_1\) and \(\frac{3}{2}\) for \(z_2\). Both are the same as the ratio of the constant terms \(\frac{8}{4}=2\). Hence, both equations are proportional.

Step 3: Analysis of Linearity

Since the equations are proportional, they represent the same line in a two-dimensional plane. This means they do not intersect each other at a unique point but in fact coincide.

Step 4: Conclusion about Solutions

Since the equations represent the same line, the system of equations does not have a unique solution. Instead, it has infinitely many solutions, corresponding to all points on the line \(6z_1 + 3z_2 = 8\).

Key Concepts

Unique Solution

In a system of equations, a "unique solution" means there is only one set of values for the variables that satisfies all the equations simultaneously. It happens when the lines or planes described by the equations intersect at exactly one point. If a system of linear equations is in two variables, this unique solution is represented graphically by the intersecting point of two lines. To determine whether a unique solution exists, it's essential to assess the independence of each equation in the system.

There are several scenarios related to solutions in linear systems:

• A unique solution exists if and only if none of the equations in the system are proportional to one another.
• If all equations are proportional, they are essentially the same, leading to infinitely many solutions.
• In the case where equations describe parallel but distinct lines, there are no solutions as the lines never meet.

Understanding these concepts can clarify when a unique solution is present and when it is not.

Proportional Equations

When two equations are proportional, they essentially describe the same line in a graphical representation. Proportionality in equations means that each term in one equation can be converted to the corresponding term in another equation by multiplying by a constant factor. This includes both the coefficients of the variables and the constant term on the other side of the equation. If all parts of one equation have a common factor with another, then these two equations are proportional.

Consider the exercises given:
The equations are: 1. \(6z_1 + 3z_2 = 8\) 2. \(4z_1 + 2z_2 = 4\) To check if they are proportional, we find that the ratios of the coefficients of \(z_1\) and \(z_2\) are both \(\frac{3}{2}\). The ratio of constant terms is \(2\), which matches these proportions. Hence, both equations represent the same line.

This means, there isn't a single point of intersection, in fact, all points on the line are solutions, leading to infinitely many solutions rather than a unique one.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations. It provides essential tools for understanding systems of equations and their solutions. Key concepts in linear algebra, such as matrices and determinants, allow us to methodically solve systems with multiple variables.

In the context of solving systems, linear algebra helps by providing techniques to:

• Determine the number of solutions via the rank of a matrix.
• Analyze line intersections in multiple dimensions using vector spaces.
• Understand whether a unique solution, no solution, or infinitely many solutions exist based on the structure of the coefficient matrix.

For example, in a two-equation system like ours, linear algebra reveals that the proportionality of equations implies they're not independent. Hence, instead of intersecting at a single point, they overlap completely.

The study of linear algebra enhances your ability to approach complex systems analytically, determining outcomes that might not be immediately obvious by observation alone.