Problem 36 - Precalculus: Mathematical 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 correct to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$\left\\{\begin{aligned} 18.72 x-14.91 y &=12.33 \\ 6.21 x-12.92 y &=17.82 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The intersection point of the two lines is \((1.14, 0.62)\).

Step 1: Rearrange the First Equation for y

Start with the equation \( 18.72x - 14.91y = 12.33 \). To solve for \( y \), isolate it on one side of the equation:\( -14.91y = -18.72x + 12.33 \).Divide every term by -14.91:\[ y = \frac{18.72}{14.91}x - \frac{12.33}{14.91} \].This simplifies to \( y \approx 1.2560x - 0.8265 \).

Step 2: Rearrange the Second Equation for y

Now, take the second equation \( 6.21x - 12.92y = 17.82 \). Similarly, solve for \( y \).\( -12.92y = -6.21x + 17.82 \).Divide each term by -12.92:\[ y = \frac{6.21}{12.92}x - \frac{17.82}{12.92} \].This simplifies to \( y \approx 0.4806x - 1.3790 \).

Step 3: Set Up a Graphing Device

Use a graphing calculator or software to input both equations. Enter the functions based on your rearrangements:- \( y = 1.2560x - 0.8265 \)- \( y = 0.4806x - 1.3790 \).Ensure both are in the same viewing window.

Step 4: Graph the Equations

Graph both lines using the graphing device. Adjust the viewing window if necessary so that you can clearly see where the two lines intersect.

Step 5: Find the Intersection Point

Use the `Intersect` function of the graphing calculator. Move the cursor close to where the two lines intersect and follow the device's instructions to find the intersection point. You can refine this by using the `TRACE` function to get an accurate reading.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions and solving equations through graphs. When dealing with linear equations, you first need to rearrange them so that they are written as functions of \( y \). This is especially important for graphing calculators because they require equations to be in the form \( y = mx + b \) to plot them correctly. After rearranging, you enter these equations into your calculator.

By setting both functions in the same viewing rectangle, you ensure the graphs are plotted on the same axes, which is crucial for identifying any points of intersection. If the intersection point isn't visible initially, adjusting the viewing window or using features like zoom can help.

Graphing calculators often come with options like `TRACE` and `Intersect`, which allow for precise navigation through the graph. `TRACE` lets you move along the graph line by line, examining point values, whereas `Intersect` locates precisely where two lines cross. This makes them particularly useful for finding solutions to systems of equations graphically.

Intersection of Lines

The intersection of lines is a key concept in solving systems of linear equations graphically. When two lines are plotted on a graph, their intersection point represents the set of values \( (x, y) \) that satisfy both equations. Finding this point graphically can give a visual understanding of the equations' solution.

In a graphing calculator, once the lines are plotted, the `Intersect` function helps pinpoint the exact coordinates of this point. This method not only marks the solution but also gives insight into the relationship between the equations. If the lines intersect at a single point, it indicates a unique solution. If the lines are parallel, there are no intersection points, indicating no solutions to the system. If the lines overlap completely, they share infinite solutions. Using these insights, you can determine the nature of the solutions available for your equations.

Solving Systems of Equations

Solving systems of equations involves finding the variable values that satisfy all the equations involved. When dealing with linear equations, this can be achieved visually through graphing. By plotting each equation as a line on a graph, the solution to the system corresponds to the intersection point of these lines.

It's crucial to first express each equation in the slope-intercept form \( y = mx + b \) to facilitate graphing. Once graphed, different methods, such as the `Intersect` or `Zoom` functions, allow you to locate the intersection. This graphical solution is often accurate for decimal places, as required.

• When lines intersect at a point, the system has a unique solution.
• If lines are parallel and distinct, the system has no solution.
• If lines coincide, there are infinitely many solutions.

Using a graphing method not only provides the solution but also enhances understanding of the equations' graphical relationships, offering a more intuitive grasp of the concept.