Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the system. $$ \left\\{\begin{aligned} 3 m-4 n &=2 \\ -6 m+8 n &=-4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions, described by \( (m, n) = \left(\frac{4n+2}{3}, n \right) \).

Step 1: Inspect the Equations

Look at both equations given in the system. The system is: \( 3m - 4n = 2 \) and \( -6m + 8n = -4 \). Notice that the second equation is a multiple of the first equation.

Step 2: Recognize the Dependent System

Observe that multiplying the first equation by \(-2\) results in the second equation. Thus, they represent the same line, indicating infinitely many solutions exist.

Step 3: Find a Parametric Solution

Express one variable in terms of the other to find a parametric solution. From the first equation, \( m = \frac{4n + 2}{3} \). Each combination of \( n \) and \( m = \frac{4n + 2}{3} \) satisfies both equations.

Key Concepts

Dependent System

A dependent system of linear equations occurs when each equation in the system represents the same line in a coordinate plane. This means the lines are not just parallel, but **coincident**—they lie on top of each other.
For example, in the system:

• \(3m - 4n = 2\)
• \(-6m + 8n = -4\)

The second equation can be obtained by multiplying the first equation by \(-2\). Hence, they are not independent, but dependent on each other. This **dependency** implies that every solution of the first equation will also satisfy the second equation.
Dependent systems are crucial concepts in linear algebra because they highlight how sometimes equations may not provide unique solutions but instead reveal relationships between variables.

Infinitely Many Solutions

When you have a dependent system of equations, like the one in our example, you will find that there are infinitely many solutions. This means that there isn't just one unique solution, but countless solutions that all satisfy both equations in the system.
This infinite number of solutions arises because both equations represent the same line. Thus, any point on that line is a solution to the system. In our example:

• Every combination of \(m\) and \(n\) that lies on the line described by \(3m - 4n = 2\) satisfies both equations.

Knowing there are infinitely many solutions is essential when analyzing real-world problems with constraints that do not bind variables to a single value.
It allows for a vast range of possibilities and shows the interconnected nature of the variables involved.

Parametric Solution

A parametric solution involves expressing one or more variables in terms of others within the system, using a parameter like \(t\) to represent the entire set of solutions. This is especially valuable for systems with infinitely many solutions.
In our example, to find such a solution, you can express the solution in terms of \(n\),

• From \(3m - 4n = 2\), solving for \(m\) gives \(m = \frac{4n + 2}{3}\).

By choosing different values for \(n\), you can find corresponding values for \(m\), utilizing the formula to cover every possible solution to the system.
A parametric solution is practical as it provides a framework for understanding and generating solutions, which can be particularly useful in computational methods and simulations in various applications.