Problem 37 - Algebra and Trigonometry

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 correct to two decimal places, either by zooming in and using TRACE 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

The solution is approximately \((x, y) = (20.70, 2.88)\).

Step 1: Solve for y in the first equation

Rewrite the first equation \(2371x - 6552y = 13,591\) to solve for \(y\).Add \(6552y\) to both sides getting \(2371x = 6552y + 13,591\).Subtract \(13,591\) from both sides to isolate terms with \(y\): \(2371x - 13,591 = 6552y\).Finally, divide by \(6552\): \(y = \frac{2371}{6552}x - \frac{13,591}{6552}\).

Step 2: Solve for y in the second equation

Rewrite the second equation \(9815x + 992y = 618,555\) to solve for \(y\).Subtract \(9815x\) from both sides: \(992y = -9815x + 618,555\).Divide by \(992\): \(y = -\frac{9815}{992}x + \frac{618,555}{992}\).

Step 3: Graph both equations

Using a graphing calculator or device, enter the two equations:1. \(y = \frac{2371}{6552}x - \frac{13591}{6552}\).2. \(y = -\frac{9815}{992}x + \frac{618555}{992}\).Make sure both lines are plotted within the same viewing rectangle to observe their intersection.

Step 4: Find the Intersection Point

Use the graphing calculator's "Intersect" feature to find the intersection point of the two lines. Adjust the viewing window if necessary to accurately locate the intersection. Note the \((x, y)\) coordinate of the point where the two lines intersect.

Step 5: Record the Solution

Once the intersection point is found, round the \(x\) and \(y\) coordinates to two decimal places to get the approximate solution to the system.

Key Concepts

Solving Linear Equations

To solve linear equations, we need to express one variable in terms of the other. When dealing with systems of equations, like two linear equations, this technique is particularly useful for graphing. In our provided equations, we identify the variable to isolate—usually, it's easier to solve for \(y\) in terms of \(x\).

For instance, consider the equation \(2371x - 6552y = 13,591\). By rearranging it, we aim to have \(y\) on one side.

Here's how you can transform it step-by-step:

• Add \(6552y\) to both sides to begin isolating \(y\).
• Subtract \(13,591\) from both sides to group all terms with \(y\) on one side.
• Divide by the coefficient of \(y\) (in this case, \(6552\)) to finalize the equation as \(y = \frac{2371}{6552}x - \frac{13,591}{6552}\).

Repeating similar steps on the second equation ensures both are in a suitable form for graphing.

Graphing Calculator

A graphing calculator is a potent tool for visualizing equations, especially complex systems. Once equations are in the \(y = mx + b\) format, inputting them becomes straightforward, providing an immediate graphical representation.

The steps to graph using a calculator involve entering each solved equation individually. This allows you to see the lines every equation represents on a single graph.

Here’s what you do:

• Turn on your graphing calculator and access the graphing tool.
• Enter the first equation resolved for \(y\).
• Enter the second equation similarly.
• Ensure both plots display on the same graph or viewing rectangle.

This visualization helps in identifying the point where the two lines meet, which is crucial in solving the system of equations.

Intersection of Two Lines

The intersection of two lines stemming from two equations provides the solution to the system of equations. It is the point that satisfies both equations simultaneously—the values of \(x\) and \(y\) where the two lines cross.

To locate this intersection precisely using a graphing calculator, utilize features like "Trace" or "Intersect." Here's how you can do it:

• On the graph display, check where the lines intersect visually.
• Use the calculator’s "Intersect" function if available. This feature automatically calculates the point where both lines meet.
• If using "Trace," manually follow the line until they converge, noting down the coordinates displayed.

The intersection signifies the solution to the equation system, providing exact values for both \(x\) and \(y\).

Decimal Approximation

Decimal approximation is essential for practical solutions. After identifying the intersection, rounding the coordinates to two decimal points renders them more manageable and fits standard precision requirements in calculations.

Working with decimals involves recognizing that exact values might be challenging due to irrational numbers or very lengthy decimals. Either way, maintaining consistency and preciseness in rounding helps.

When you find the intersection on the graph:

• Carefully note the \((x, y)\) values displayed.
• Round each value to two decimal places, ensuring clarity and correctness.
• Document these values as they represent the approximate solution to your system of equations.

Decimal approximations are critical for both classroom solutions and real-world applications, balancing exactness and simplicity.