Problem 108 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: \(\left(\frac{3}{4}, 0\right)\) \(y\) -intercept: \(\left(0, \frac{4}{3}\right)\)

Step-by-Step Solution

Answer

The equation of the line that passes through the points \(\left(\frac{3}{4}, 0\right)\) and \(\left(0, \frac{4}{3}\right)\) is \(y = - \frac{16}{9}x + \frac{4}{3}\).

Step 1: Compute The Slope

Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), and replacing \(x_1, y_1\) with the coordinates of the x-intercept \(\left(\frac{3}{4}, 0\right)\) and \(x_2, y_2\) with the coordinates of the y-intercept \(\left(0, \frac{4}{3}\right)\) , we can compute the slope as follows: \(m = \frac{\frac{4}{3} - 0}{0 - \frac{3}{4}} = - \frac{16}{9}\)

Step 2: Find The Equation of The Line

Now that we have calculated the slope \(m\) and we have the y-intercept \(b\), we can substitute them into the line equation \(y = mx + b\). Here \(m = - \frac{16}{9}\) and \(b = \frac{4}{3}\), thus the equation of the line is: \(y = - \frac{16}{9}x + \frac{4}{3}\).

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and is an essential aspect of understanding linear equations. To find the slope, or gradient, between two points, one can use the slope formula, which is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Consider two points on a graph, Point 1 with coordinates \((x_1, y_1)\) and Point 2 with coordinates \((x_2, y_2)\). The difference in the y-coordinates (vertical change) is divided by the difference in the x-coordinates (horizontal change) to determine the slope, which can be positive, negative, zero, or undefined. A positive slope means the line ascends from left to right, a negative slope means it descends, a zero slope indicates a horizontal line, and an undefined slope corresponds to a vertical line.

In the exercise provided, the slope is calculated between the x-intercept and the y-intercept. The x-intercept is the point where the line crosses the x-axis, and similarly, the y-intercept is where the line crosses the y-axis. Substituting the given intercepts into the slope formula gives us the slope of the line, leading us to the next steps in defining the line's equation.

X-Intercept and Y-Intercept

In the context of linear equations, intercepts are specific points where the line crosses the axes. The x-intercept is the point at which the line intersects the x-axis, and at this point, the y-value is zero. Conversely, the y-intercept is where the line meets the y-axis, making the x-value zero at this junction.

To identify these points on a graph, one only needs to check where the plotted line cuts through the axes. These intercepts are particularly useful as they provide two distinct points through which the line passes and can be pivotal in formulating the equation of the line, as we've seen in the exercise solution. When the coordinates of these intercepts are plugged into the slope formula, they assist in finding the slope, a necessary step before writing the line's equation in slope-intercept form, which forms an integral part of plotting and understanding linear equations in geometry and algebra.

Slope-Intercept Form

The slope-intercept form is one of the most straightforward ways to express the equation of a straight line. This form is written as \( y = mx + b \), where \( m \) denotes the slope of the line, and \( b \) represents the y-intercept. It's a popular form because it provides immediate visual information about the line's slope and where it intersects the y-axis.

Understanding Slope-Intercept Form

The value of \( m \) tells us how steep the line is, and its sign indicates the direction the line tilts. The y-intercept, \( b \), then gives us a starting point to plot the line on the graph. Once the slope and y-intercept are known, the line can be easily drawn by starting at \( b \) on the y-axis and using \( m \) to determine the angle of ascent or descent.

Reflecting on the original exercise, by using the slope \( m = - \frac{16}{9} \) that was calculated using the slope formula and the given y-intercept \( b = \frac{4}{3} \), we successfully crafted the slope-intercept form equation of the line. This particular equation helps in visualizing the line on a graph and is crucial for further analysis in various applications such as physics, economics, and statistics.