Problem 1 - Technical Mathematics with Calculus

Question

Graphically find the approximate solution to each system of equations. If you have a graphics calculator, use the \([\mathrm{ZOOM}]\) and \(\mathrm{TRACE}\). or \([\text { INTERSECT }]\) features to find the solution. $$\begin{aligned} &2 x-y=5\\\ &x-3 y=5 \end{aligned}$$

Step-by-Step Solution

Answer

By graphing the two equations, the approximate solution to the system is found at the intersection point of the two lines.

Step 1 - Graph the first equation

Begin by rearranging the first equation, \(2x - y = 5\), to solve for \(y\) in terms of \(x\). This gives \(y = 2x - 5\). Plot this equation on a graph or use a graphics calculator to represent this linear equation.

Step 2 - Graph the second equation

Next, rearrange the second equation, \(x - 3y = 5\), to solve for \(y\) in terms of \(x\). The rearranged form will be \(y = \frac{1}{3}x - \frac{5}{3}\). Plot this equation on the graph as well, or input it into the graphics calculator, ensuring the line is distinct from the first.

Step 3 - Identify the point of intersection

Using the graph, observe where the two lines intersect. This intersection point is the solution to the system of equations. If using a graphics calculator, you can utilize the Zoom or Trace feature to better view the intersection, or the Intersect feature to find the exact coordinates of the intersection point.

Step 4 - Approximate the solution

After identifying the intersection point either graphically or with a calculator's intersection feature, record the coordinates as your approximate solution to the system of equations.

Key Concepts

Graphical Solution Method

The graphical solution method is a visual approach to solving systems of equations. It involves plotting the graphs of all the equations in the system and then identifying where they intersect. This method is particularly useful for systems of linear equations, where the graphs are straight lines.
In our case, we've been given a system of two equations:

1. For the first equation, we rearrange it to get the slope-intercept form, which is: \(y = 2x - 5\).
2. For the second equation, we also rearrange to get \(y = \frac{1}{3}x - \frac{5}{3}\).

Each equation represents a line on a coordinate plane. The solution to the system of equations is the point where the two lines intersect. By graphing both lines on the same set of axes, we can visually identify the approximate point of intersection.

Graphics Calculator Usage

A graphics calculator can be an extremely handy tool when solving systems of equations graphically. Once you have entered the equations into the calculator in their graphical form, you can use powerful features to assist you.

1. The \([\mathrm{ZOOM}]\) feature can help you to get a better view of the area where the lines may intersect.
2. The \(\mathrm{TRACE}\) function allows you to move along the curves to find the point of intersection with greater precision.
3. Lastly, the \([\text{INTERSECT}]\) feature can be used to directly calculate and display the intersection point on the screen.

It's important to become familiar with your calculator's capabilities, as these features can save much time and increase accuracy, offering more exact solutions than manual graphing.

Linear Equation Graphing

Graphing linear equations is the foundation of the graphical solution method. Each linear equation in a two-variable form (like \(x\) and \(y\)) can be transformed into a graph on a two-dimensional plane.
To graph a linear equation, you need to convert it into the slope-intercept form \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. This form makes it easy to plot the line on a graph by:

1. Start plotting the y-intercept \(b\) on the y-axis.
2. Use the slope \(m\) to find another point on the line by 'rising' and 'running' from the intercept. 'Rise' refers to the movement upward (positive slope) or downward (negative slope), and 'run' refers to the movement to the right along the x-axis. For a fraction slope like \(\frac{1}{3}\), you rise 1 unit and run 3 units.

Repeat this process for all the equations in your system, and you will have a graph with lines that represents each equation.

Intersection Point Identification

Once the linear equations have been graphed, the next step is to identify the intersection point. This point, at which the lines cross, represents the solution to the system of equations, meaning it is the set of coordinates \( (x, y) \) that satisfies all the equations simultaneously.
To identify this point on a graph, look for the location where the lines intersect and note the coordinates. Accuracy matters here, as an incorrect intersection point will lead to an incorrect solution. If you're using graph paper, it's possible to estimate this visually; however, a graphics calculator provides tools to pinpoint this intersection more accurately. Understanding how to read the graph correctly and identify the intersection can be critical in correctly solving the system of equations.