Problem 51 - 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 rounded to two decimal places, either by zooming in and using [RACE ] or by using Intersect. $$\left\\{\begin{array}{l}2371 x-6552 y=13,591 \\\9815 x+992 y=618,555\end{array}\right.$$

Step-by-Step Solution

Answer

Solve for \(y\), graph both lines, find, and round the intersection point.

Step 1: Solve for y in the first equation

Starting with the equation \(2371x - 6552y = 13591\), we want to express \(y\) in terms of \(x\). First, subtract \(2371x\) from both sides to get \(-6552y = -2371x + 13591\). Then, divide the entire equation by \(-6552\) to isolate \(y\):\[y = \frac{2371}{6552}x - \frac{13591}{6552}\].

Step 2: Solve for y in the second equation

For the second equation \(9815x + 992y = 618555\), we need to isolate \(y\) as well. Start by subtracting \(9815x\) from both sides, resulting in \(992y = -9815x + 618555\). Next, divide the entire equation by \(992\) to solve for \(y\):\[y = -\frac{9815}{992}x + \frac{618555}{992}\].

Step 3: Graph both equations

On a graphing device, input the expressions for \(y\) from Step 1 and Step 2. Graph these two lines in the same viewing window to visualize their intersection. This can help us determine the approximate solution of the system of equations.

Step 4: Find the intersection (using Intersect)

After graphing, use the Intersect function on the graphing device to calculate where the two lines intersect. This intersection point \((x, y)\) represents the solution to the system of equations.

Step 5: Round the solution

Ensure that the resulting intersection point \((x, y)\) from Step 4 is rounded to two decimal places to meet the problem's requirement. This yields the final solution for the system of equations.

Key Concepts

Solve for Y

Before we can graph the system of equations, it's necessary to rewrite each equation so that both are in the slope-intercept form (or "solve for y"). This means we need equations that have "y" on one side and an expression involving "x" on the other. For example, consider the first equation: \(2371x - 6552y = 13591\). We start by isolating \(y\) by moving all terms involving \(x\) to the opposite side, resulting in \(-6552y = -2371x + 13591\). Dividing through by \(-6552\), we isolate \(y\):

• \(y = \frac{2371}{6552}x - \frac{13591}{6552}\)

For the second equation, \(9815x + 992y = 618555\), use a similar process: move \(x\) to the other side and divide by \(992\) to get:

• \(y = -\frac{9815}{992}x + \frac{618555}{992}\)

Once both equations are in this form, they're ready to be input into a graphing calculator. This is a crucial step because a calculator that graphs will need a clear input for "y" in terms of "x".

Graphing Calculator

When you want to solve a system of equations using a graphing calculator, it's essential first to convert the equations into a graphable format. A graphing calculator is a valuable tool because it simplifies the process of plotting complex equations and visualizing their intersections. To input the equations, you have precisely isolated \(y\) for both equations, turning them into slope-intercept form.
After entering these into the calculator:

• Make sure both equations are graphed in the same viewing window, allowing easy comparison and intersection observation.
• Graph the equations by selecting the corresponding graphing function, which will automatically draw the lines based on the inputs provided.

Ensure the viewing window is adjusted correctly to see where the lines might intersect. This adjustment might require zooming in or panning to find a suitable area, so both lines appear clearly.

Intersection of Lines

Once the equations are graphed, finding the point where the lines intersect is where the beauty of algebra meets visual representation. This intersection point represents the solution to the system of equations, the values of \(x\) and \(y\) that satisfy both equations.
There are a few methods to find this intersection using your graphing calculator:

• Utilize the "Intersect" function. This tool is designed to find and display the exact point where the two lines meet.
• If needed, you can zoom into the intersection area to pinpoint the solution more easily. This method is particularly helpful if the intersection is not initially clear.

After identifying the intersection point, make sure to round \((x, y)\) to two decimal places for the final answer. This process clearly outlines the system's solution, providing a practical application of both algebra and technology.