Problem 118 - College Algebra Essentials

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 x-coordinate of the point where \(y_1\) and \(y_2\) intersect is the solution to the equation.

Step 1: Set up \(y_1\) and \(y_2\)

First, separate the equation into two parts and assign to \(y_{1}\) and \(y_{2}\) respectively. Therefore, \(y_{1}=\frac{x-3}{5}-1\) and \(y_{2}=\frac{x-5}{4}\). Each of these is an equation for a line with slope and y-intercept.

Step 2: Graph \(y_1\) and \(y_2\)

Next, input \(y_{1}\) and \(y_{2}\) into the graphing utility, and create the graph. This will result in two linear functions displayed on the graph.

Step 3: Find the intersection

Observe the graph and locate the point where \(y_{1}\) and \(y_{2}\) intersect. This point represents the solution to the equation - the value of the x-coordinate is the solution since at that point, both \(y_1\) and \(y_2\) expressions are equal.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps you visualize mathematical functions and equations. This can be a physical calculator or a digital application. It allows you to input equations and see their corresponding graphs. These visual representations are incredibly useful when solving problems. For instance, when you deal with complex equations, graphing utilities can display the solutions quickly and accurately.

To use a graphing utility effectively, you need to enter each part of your equation as separate functions. In the example given, this means treating each side of the equation as its own function. Once entered, the utility can provide either a graph or a table of values. This helps you quickly identify solutions.

Intersection of Lines

The intersection of lines occurs where two lines on a graph cross one another. In solving linear equations, this intersection represents the solution to the equation. This is because at the intersection point, the values for the x and y coordinates satisfy both of the original equations.

In the original exercise, the lines represented by the equations of \(y_1\) and \(y_2\) intersect. The x-coordinate of this intersection point gives the solution to the equation. It's critical because it visually demonstrates that the left side and right side of the equation are equal at this point. Knowing how to find and interpret intersections is vital for solving many types of equations.

Linear Functions

Linear functions are mathematical expressions that create straight lines when graphed. They have the general form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept. These functions are foundational in algebra and are encountered frequently in equations.

Each side of the original equation creates its own linear function. Understanding the properties of linear functions, such as slope and intercept, assists in predicting how the graph will look and behave. For instance, different slopes indicate different angles, and different intercepts suggest that lines will cross the y-axis at different points. Mastery of these concepts allows for quicker and more intuitive graphing.

Graphing Equations

Graphing equations involves plotting their solutions on a coordinate plane. Through graphing, you can visually analyze the relationship between variables and the behavior of the function. This method is especially useful for finding points of intersection, roots, and general trends in data.

In practice, an equation like \(\frac{x-3}{5}-1 = \frac{x-5}{4}\) would be graphed as two separate lines using each side of the equation. By plotting these on the same axes, their point of intersection becomes apparent, revealing the solution. Graphing not only offers a solution but also provides visual context and understanding, making it an essential skill in mathematics.