Problem 4 - Calculus for Biology and Medicine

Question

Solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{array}{l} 2 x+y=\frac{1}{3} \\ 6 x+3 y=1 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions; both equations represent the same line.

Step 1: Write the System of Equations

The given system of equations is: \[ \begin{align*} 2x + y &= \frac{1}{3} \ 6x + 3y &= 1 \end{align*} \] Here, we have two equations that we need to solve simultaneously.

Step 2: Simplify the Second Equation

The second equation can be simplified. Divide every term by 3: \[ 6x + 3y = 1 \] \[ \frac{6x}{3} + \frac{3y}{3} = \frac{1}{3} \] This simplifies to: \[ 2x + y = \frac{1}{3} \] Notice that both equations are now the same: \[ 2x + y = \frac{1}{3} \] \[ 2x + y = \frac{1}{3} \] This indicates that the system has infinitely many solutions.

Step 3: Analyze the System of Equations

Since both equations are identical, the system is dependent. This means any point on the line \(2x + y = \frac{1}{3}\) is a solution, resulting in infinitely many solutions.

Step 4: Graph Both Equations

As both equations are the same, they will graph as a single line. To graph \(2x + y = \frac{1}{3}\), plot at least two points. For example: when \(x = 0\), \(y = \frac{1}{3}\), giving point \( (0, \frac{1}{3})\). When \(y = 0\), \(x = \frac{1}{6}\), giving point \( (\frac{1}{6}, 0)\). Draw a line through these points.

Step 5: Conclude Based on the Graph

The graph is a single straight line since both given equations are the same. This confirms that every point on this line satisfies both equations, reconfirming the system has infinitely many solutions.

Key Concepts

Graphical Solution

In mathematics, solving a system of linear equations using a graphical solution involves plotting each equation on a coordinate plane and identifying points where they intersect. For this particular exercise, we had two equations:

• First Equation: \(2x + y = \frac{1}{3}\)
• Second Equation: \(6x + 3y = 1\)

To solve graphically, both equations were plotted. However, upon simplifying the second equation, it was revealed that it is the same as the first equation. This means when plotted, both equations will appear as a single line.
When graphing, selecting points like \((0, \frac{1}{3})\) and \((\frac{1}{6}, 0)\) allows the formation of the graph. By drawing a line through these points, you visually represent the solutions.
Graphing is a visual way to confirm solutions. If different lines intersect at a point, that's the solution. But here, the overlapping lines indicate a different kind of solution: infinite solutions.

Dependent System

A dependent system arises when two equations describe the same line or curve. They are not distinct because one equation is a multiple or a manipulation of the other. In this exercise, after simplifying, both equations turned out to be identical:

• \(2x + y = \frac{1}{3}\)
• \(6x + 3y = 1\), simplified to \(2x + y = \frac{1}{3}\)

This shows that there was no distinct intersection point to find. Instead, it’s clear that every point on \(2x + y = \frac{1}{3}\) line satisfies both original equations.
Dependent systems are important because they show situations where multiple representations lead to the same outcomes. Recognizing a dependent system saves time and provides insight that we’re dealing with overlapping equations on a graph.

Infinite Solutions

When a system of equations has infinite solutions, it means there's not just one or a finite number of solutions, but an entire collection of them. This occurs when the equations represent the same line. In this exercise, analyzing the solved system revealed:

• The equations \(2x + y = \frac{1}{3}\) and \(6x + 3y = 1\) coincide after the second one is simplified.

This overlap signifies infinitely many solutions, as any point along the line is a solution.
Rather than a single answer, infinite solutions suggest numerous options satisfying both equations. This is shown graphically by the single line that both equations plot to, providing clarity that any coordinate along this line effectively serves as a solution. Understanding infinite solutions broadens our view of linear systems, emphasizing that equations sometimes define endless possibilities rather than a single outcome.