Problem 118 - Algebra and Trigonometry

Question

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2}\). Then use the utility's [TABLE] or [ GRAPH] feature to solve the equation. $$\frac{x-3}{5}-1=\frac{x-5}{4}$$

Step-by-Step Solution

Answer

The solution to the equation is the x-coordinate of the point where the two graphs intersect. This value can be found by carefully examining the graph or the table of values on the graphing utility.

Step 1: Setting Up the Functions

The equation given is \(\frac{x−3}{5}−1=\frac{x−5}{4}\). This can be separated into two different functions, that will be entered into the graphing utility: \(y_{1} = \frac{x−3}{5}−1\) and \(y_{2} = \frac{x−5}{4}\)

Step 2: Graphing the Functions

Enter each function, \(y_{1}\) and \(y_{2}\), into the graphing utility, and create the corresponding graphs. Ensure both graphs are visible on the screen at the same time.

Step 3: Finding the Point of Intersection

Observe the graph or the table of values on the graphing utility to find where \(y_{1}=y_{2}\), which is the solution to the original equation. The x-coordinate at this point will be the solution.

Key Concepts

Functions Intersection

When solving an equation graphically, understanding the concept of the intersection of functions is critical. In our case, each side of the given equation forms a separate function:

• Function 1: \( y_{1} = \frac{x-3}{5} - 1 \)
• Function 2: \( y_{2} = \frac{x-5}{4} \)

These functions, when plotted in a graphing utility, create two lines. The point where these lines intersect represents the solution to the equation, because at this point both functions have the same y-value for the same x-value. This means that their expressions are equal, just as the original equation suggests. Understanding intersections in this way helps visualize why the x-coordinate at the point of intersection solves the equation. Thus, the task is reduced to finding where these two lines meet on the graph.

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions. It's like a calculator, but instead of just numbers, it handles and displays graphs. When dealing with equations such as \( \frac{x-3}{5}-1=\frac{x-5}{4} \), using a graphing utility simplifies the process by visualizing the functions as lines or curves.

• You enter the function expressions, like \( y_{1} \) and \( y_{2} \), into the graphing utility.
• The utility plots these functions on a coordinate grid.

You can compare the graphs visually, either using the screen to look for intersection points or using specific features like tables or intersection finders, providing an intuitive way to solve complex equations. Notably, the visual component enhances understanding, making it easier to interpret equation solutions.

Solving Equations Graphically

Solving equations graphically is a method where you use graphs instead of algebraic manipulation to find solutions. This is particularly useful for more complex equations that are difficult to rearrange or simplify. In our example, each side of the equation is treated as a separate function. These are graphed using a utility, which allows you to pinpoint where the graphs intersect. Here’s how graphing can solve equations:

• Each equation side is treated as its own function and plotted separately.
• The solution to the equation corresponds to the x-coordinate at the intersection of the graphs.

This method is both visual and hands-on, providing a contrast to purely symbolic methods of solving equations. This approach is especially beneficial when equations are non-linear or complex, giving students and mathematicians an alternative tool to approach problem-solving.