Problem 4 - Calculus for Biology and Medicine

Question

I n ~ P r o b l e m s ~ \(1-4\), 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 as both equations represent the same line.

Step 1: Simplify the System of Equations

The given system of equations is:\[\begin{array}{l}2x + y = \frac{1}{3} \6x + 3y = 1\end{array}\]Notice that the second equation can be reduced by dividing all terms by 3:\[2x + y = \frac{1}{3}\]Now the system looks as follows:\[\begin{array}{l}2x + y = \frac{1}{3} \2x + y = \frac{1}{3}\end{array}\]

Step 2: Identify the Nature of the System

Both equations are identical after simplification. This means that every point on the line \(2x + y = \frac{1}{3}\) is a solution to the system. Therefore, there are infinitely many solutions. The lines coincide on the graph.

Step 3: Plot the Graph of the System

To plot the line, rewrite the equation in slope-intercept form (\(y = mx + b\)). For the equation \(2x + y = \frac{1}{3}\), solve for \(y\):\[y = -2x + \frac{1}{3}\]This line has a slope of \(-2\) and a y-intercept of \(\frac{1}{3}\). When you plot it, ensure that every point on this line is a solution to the system, indicating infinite solutions.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables, which you solve together to find a common solution. In the exercise provided, we are dealing with a system consisting of two linear equations:

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

The objective is to find values for \(x\) and \(y\) that satisfy both equations simultaneously. Often, systems of equations are solved using substitution, elimination, or graphing methods.
In this case, we simplified the equations by noticing the second equation could be divided by 3. This simplification revealed that both equations were, in fact, the same. Thus, any solution to one equation automatically solves the other, showing the interconnection in the system.

Graphing Linear Equations

Graphing linear equations is one of the most intuitive ways to solve a system of equations. It involves drawing the lines represented by each equation and finding their point of intersection. In simpler systems, this point provides the solution for the variables involved.
To graph the equation \(2x + y = \frac{1}{3}\), you first convert it into slope-intercept form, \(y = mx + b\). Here, it becomes \(y = -2x + \frac{1}{3}\).

• The slope \(m = -2\)
• The y-intercept \(b = \frac{1}{3}\)

The slope, \(-2\), tells us that for each unit increase in \(x\), \(y\) decreases by 2 units. The y-intercept is the point where the line crosses the y-axis. Once plotted, the graph of the equation is a straight line where every point represents a solution. Because both equations are the same line, they illustrate that all points on the line are solutions to the system.

Infinitely Many Solutions

In some systems of equations, such as the one in the exercise, you might find that there are infinitely many solutions. This occurs when both equations represent the same line and overlap entirely on the graph.
When you simplify the given system of equations and find both to be identical, like:

• \(2x + y = \frac{1}{3}\)
• \(2x + y = \frac{1}{3}\)

It becomes evident that any point on the line \(y = -2x + \frac{1}{3}\) is a solution. This is a characteristic situation in linear systems where the lines don't cross at just one point but rather coincide fully.
Thus, understanding that infinitely many solutions signify overlapping lines helps visually confirm the equality of the system's equations on a graph.