Problem 82 - Introductory Algebra for College Students

Question

Use a graphing utility to solve each rational equation. Graph each side of the equation in the given viewing rectangle. The first coordinate of each point of intersection is a solution. Check by direct substitution. $$\begin{aligned} &\frac{50}{x}=2 x\\\ &[-10,10,1] \text { by }[-20,20,2] \end{aligned}$$

Step-by-Step Solution

Answer

The solution of the equation will be the x-coordinate of the intersection points of the graphs \(y=\frac{50}{x}\) and \(y=2x\), confirmed by substituting each solution back into the original equation.

Step 1: Graph Rational Function and Linear Function

Using a graphing utility, graph the rational function \(y = \frac{50}{x}\) and the linear function \(y = 2x\) in the viewing rectangle [-10,10,1] by [-20,20,2]. Both the functions will appear as continuous lines on the graph.

Step 2: Identify Points of Intersection

Look for any points where the graphs of the two functions intersect. The x-coordinate of each point of intersection is a solution to the equation. Here, there might be multiple points of intersection, so we must consider all of them.

Step 3: Check Solutions by Direct Substitution

Substitute each x-value obtained in Step 2 into the original equation \( \frac{50}{x} = 2x \) to see if both sides of the equation are equal. If they are equal, then the solution is correct. If not, check the work in the previous steps.

Key Concepts

Graphing Utility

Graphing utilities are tools that help us visually interpret mathematical equations on a graph. They are incredibly useful when dealing with equations that may be difficult to solve algebraically. For this exercise, a graphing utility is used to graph the functions \(y = \frac{50}{x}\) and \(y = 2x\).
By plotting both functions on a graph, you can easily see how they behave and interact with one another. The viewing rectangle specified as \([-10,10,1]\) by \([-20,20,2]\) sets the range for our axes, where the x-axis ranges from -10 to 10, and the y-axis ranges from -20 to 20.

• Graphing utilities can take various forms, from handheld calculators to computer software.
• They are essential for students and professionals dealing with complex mathematical models.
• These tools make it easier to recognize patterns and relationships between different functions.

Utilizing a graphing utility is a straightforward step, and it's an important skill that enhances problem-solving efficiency.

Intersection Points

Finding intersection points is key to solving equations graphically. Once the functions are plotted, we look for points where the graphs of these functions cross each other. These points of intersection represent the solutions to the equation.
When two lines on a graph meet, the x-coordinate at this intersection is what we need. For the equation \(\frac{50}{x} = 2x\), this means that all x-coordinates at the intersection points are potential solutions. In simple terms, find where the curves meet, and that's what you're aiming for as a solution.

• The intersection points are essentially where both functions have the same y-value for a particular x-value.
• It's possible to have multiple intersection points, meaning multiple solutions.
• Even though the graph can help, always double-check the solutions to be sure.

The visual aspect of intersection points makes this method particularly effective, providing a clear indication of the solutions.

Direct Substitution

Direct substitution is the process of verifying solutions by substituting them back into the original equation. In the context of our exercise, once we have the x-values from the intersection points, we plug them back into the equation \(\frac{50}{x} = 2x\) to ensure both sides are equal.
This step is crucial because it confirms the validity of the graphical solutions obtained. Sometimes errors can occur, and direct substitution is a way to catch these mistakes.

• To use direct substitution, replace \(x\) with the x-coordinate of the intersection point.
• If both sides of the equation are equal, then it confirms the solution is correct.
• If results don't match, there might have been an error in plotting or calculation, necessitating a re-check.

Direct substitution ensures that your graphical interpretation is indeed a valid solution, providing that final check for accuracy.