Problem 53 - Precalculus: Mathematics for Calculus

Question

Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system either by zooming in and using \([\text { TRACE }]\) or by using Int er sect. Round your answers to two decimals. $$\left\\{\begin{array}{l} 2371 x-6552 y=13,591 \\ 9815 x+992 y=618,555 \end{array}\right.$$

Step-by-Step Solution

Answer

The intersection point is approximately (27.55, -3.99).

Step 1: Convert Equations to Slope-Intercept Form

To use a graphing device effectively, we must first convert both equations into the form \(y = mx + b\).**Equation 1:**\[2371x - 6552y = 13591\]Rearrange to solve for \(y\):\[6552y = 2371x - 13591\]\[y = \frac{2371}{6552}x - \frac{13591}{6552}\]**Equation 2:**\[9815x + 992y = 618555\]Rearrange to solve for \(y\):\[992y = -9815x + 618555\]\[y = -\frac{9815}{992}x + \frac{618555}{992}\]

Step 2: Graph the Equations

Now use a graphing calculator to plot the equations:**Plot 1:** \( y = \frac{2371}{6552}x - \frac{13591}{6552} \)**Plot 2:** \( y = -\frac{9815}{992}x + \frac{618555}{992} \)Ensure that both lines are visible in the viewing rectangle.

Step 3: Identify the Intersection Point

To find the point where the two lines intersect, use the [INTERSECT] function or [TRACE] option on the graphing calculator. 1. Access the [INTERSECT] function on your graphing calculator. 2. Select the two graphs and confirm the intersection. 3. The calculator will display the intersection coordinates.

Step 4: Round the Intersection Coordinates

After finding the intersection point using your graphing device, round the x and y values to two decimal places for precision. For example, if your calculator gives the intersection point as (27.5482, -3.987), round it to (27.55, -3.99).

Key Concepts

Slope-Intercept Form

When dealing with equations of a line, the slope-intercept form is incredibly useful. This form is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept.
This format makes it easy to understand how the line behaves. The slope, \(m\), indicates the direction and steepness of the line. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.
The y-intercept, \(b\), shows where the line crosses the y-axis. Converting any linear equation into slope-intercept form helps visualize and graph the line quickly, especially when using graphing tools like calculators.
In the original exercise, the equations were first rearranged into slope-intercept form to facilitate graphing. Once in this form, they can be easily plotted using a graphing calculator.

Graphing Calculator

A graphing calculator is an invaluable tool for students learning about systems of equations and graphing functions. These devices allow for visual representation of equations, making abstract algebraic expressions easier to understand.
Using a graphing calculator, you can quickly and accurately plot graphs. First, make sure your equation is in slope-intercept form (\(y = mx + b\)). Input this into the graphing calculator and access its graphing feature.
Ensure all relevant function settings are configured so that both equations appear in the visible screen or 'viewing rectangle'.

• Type each equation into a separate graph line parameter.
• Adjust the viewing window if necessary to accommodate the scope of the graphs.
• Double-check equations for input accuracy.

Once the equations are in place, the graphing calculator plots them on a set of axes, empowering you to see their intersections and relationships visually.

Intersection Point

When graphing two lines, their intersection point is where the equations hold true simultaneously. This point is the solution to the system of equations.
Finding this point visually with a graphing calculator involves using tools like the [INTERSECT] function or the [TRACE] function.
Here's how to find an intersection point with a graphing calculator:

• First, graph both lines on the calculator's display.
• Once both lines are visible, access the [INTERSECT] tool.
• Follow prompts to select the two lines you're analyzing.
• The calculator computes and displays the coordinates of the intersection.

Afterward, ensure the x and y coordinates of the intersection point are rounded to two decimal places for precision. This is a common requirement in math problems to ensure clarity and accuracy in reporting results.