Question

Write an equation of the line through the point (a) parallel to the given line and (b) perpendicular to the given line. $$ \frac{\text { Point }}{\left(\frac{3}{4}, \frac{7}{8}\right)} \quad \frac{\text { Line }}{5 x-3 y=0} $$

Short answer

The equation of the line parallel to the given line passing through the point \((3/4, 7/8)\) is \(y = \frac{5}{3}x + \frac{1}{24}\). The equation of the line perpendicular to the given line passing through the point \((3/4, 7/8)\) is \(y = -\frac{3}{5}x + \frac{41}{40}\).

Step-by-step solution

Find the slope of the given line

First, rearrange the equation of the provided line to the slope-intercept form \(y = mx + b\), where \(m\) is the slope. The line \(5x - 3y = 0\) can be rewritten as \(y = \frac{5x}{3}\). So the slope of the given line, \(m\) is \(\frac{5}{3}\).

Write equation for line parallel to the given line

The slope of a line parallel to a given line is the same as the slope of the given line. So for a line parallel to the given line, the slope is also \(\frac{5}{3}\). Use the point-slope form of a line equation with point \((3/4, 7/8)\) and slope \(\frac{5}{3}\) to write the equation \(y - \frac{7}{8} = \frac{5}{3}(x - \frac{3}{4})\). Simplifying this results in the equation of the line parallel to the original one: \(y = \frac{5}{3}x + \frac{1}{24}\).

Write equation for line perpendicular to the given line

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. The negative reciprocal of \(\frac{5}{3}\) is \(-\frac{3}{5}\). So for a line perpendicular to the given line, the slope is \(-\frac{3}{5}\). Use the point-slope form of a line equation with point \((3/4, 7/8)\) and slope \(-\frac{3}{5}\) to write the equation \(y - \frac{7}{8} = -\frac{3}{5}(x - \frac{3}{4})\). Simplifying this results in the equation of the line perpendicular to the original one: \(y = -\frac{3}{5}x + \frac{41}{40}\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for writing and graphing equations of lines. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) signifies the y-intercept, which is the point at which the line crosses the y-axis.

The beauty of the slope-intercept form lies in its simplicity for graphing. The slope tells us how steep the line is, and the y-intercept gives us a starting point on the graph. For example, if a line has an equation \(y = \frac{2}{3}x + 4\), the slope is \(\frac{2}{3}\) and the y-intercept is at \(0, 4\).

To convert a standard form equation to the slope-intercept form, you isolate \(y\) on one side of the equation. In the exercise example, \(5x - 3y = 0\) becomes \(y = \frac{5}{3}x\) once rearranged, immediately giving us the slope \(\frac{5}{3}\) and showing a y-intercept at \(0, 0\).

Point-Slope Form

The point-slope form plays a pivotal role in forming equations that pass through a specific point, and it's given by the equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \(\left(x_1, y_1\right)\) is the point through which the line passes.

This form is especially handy when you have one point and a slope and need to find the equation of the line. It allows for a direct substitute for these values without needing to first calculate the y-intercept. Following the textbook exercise, to write the equation of a line parallel to \(5x - 3y = 0\) using point \(\left(\frac{3}{4}, \frac{7}{8}\right)\), and the slope \(\frac{5}{3}\), we substitute these into point-slope form, yielding \(y - \frac{7}{8} = \frac{5}{3}(x - \frac{3}{4})\).

H4: Simplification is Key
After substituting, the next step is to simplify the equation to reach a more recognizable form, which is typically the slope-intercept form, as it was in the exercise with \(y = \frac{5}{3}x + \frac{1}{24}\).

Parallel and Perpendicular Lines

In geometry, understanding the relationship between parallel and perpendicular lines is vital. Parallel lines have identical slopes, which means if two lines are parallel, their equations' slope values will be the same. Conversely, perpendicular lines have slopes that are negative reciprocals of each other.

For instance, if one line has a slope of \(2\), a parallel line will also have a slope of \(2\), while a perpendicular line will have a slope of \(\frac{-1}{2}\). Why negative reciprocals? Because multiplying the slopes of the two perpendicular lines should yield \(m_\text{1} \cdot m_\text{2} = -1\), embodying the concept of 'opposite directions' inherent in perpendicularity.

In the exercise, once the slope of the given line \(5x - 3y = 0\) is found to be \(\frac{5}{3}\), a parallel line through a specific point will use the identical slope. However, a line perpendicular to it will use the negative reciprocal, \(\frac{-3}{5}\), as seen in the step by step solution, leading to the perpendicular line's equation \(y = -\frac{3}{5}x + \frac{41}{40}\).