Problem 88 - College Algebra and Calculus: An Applied Approach

Question

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \(\begin{array}{ll}\underline{\phantom{xxx}}\left(\frac{7}{8}, \frac{3}{4}\right) & 5 x+3 y=0\end{array}\)

Step-by-Step Solution

Answer

The equations of the lines through the point \((7/8, 3/4)\) are \[ y = -\frac{5}{3}x + \frac{85}{48} \] parallel to the given line, and \[ y = \frac{3}{5}x + \frac{3}{40} \] perpendicular to the given line.

Step 1: Find the Slope of the Given Line

First, convert the equation of the given line \(5x + 3y = 0\) to slope-intercept form \(y = mx + c\). You can do this by isolating \(y\) on one side of the equation. \[y = -\frac{5}{3}x \] This indicates that our slope for the initial line, \(m_1\), is -5/3.

Step 2: Equation of Line Parallel to the Given Line

Understanding that parallel lines keep a consistent slope, the line parallel to the initial line also has a slope of -5/3. The equation of this line (passing through the point \(7/8, 3/4\)) can be given by \[y - y1 = m(x - x1) \] Substituting for \(m\), \(x1\), and \(y1\), we get \[ y - \frac{3}{4} = -\frac{5}{3}(x - \frac{7}{8}) \] which simplifies to \[ y = -\frac{5}{3}x + \frac{85}{48} \]

Step 3: Equation of Line Perpendicular to the Given Line

A line perpendicular to another line has a slope which is the negative reciprocal of the original line's slope. Therefore the slope of the line perpendicular to the initial line, \(m_2\), is 3/5. Its equation (passing through the point \(7/8, 3/4\)) can then be written as \[ y - \frac{3}{4} = \frac{3}{5}(x - \frac{7}{8}) \] This simplifies to \[ y = \frac{3}{5}x + \frac{3}{40} \]

Key Concepts

Parallel Lines

Parallel lines are lines that never intersect or cross each other no matter how far you extend them. One of the key properties of parallel lines is that they have the same slope. This means that if you have the equation of one line in slope-intercept form, such as \( y = mx + c \), any line parallel to it will have an equation of the form \( y = mx + k \). The value of \( m \) remains the same for both, as that is the slope determining the direction of the line.

For example, if a given line has a slope of \(-\frac{5}{3}\), any line parallel to it will also have that same slope \(-\frac{5}{3}\). If you are tasked with finding such a parallel line that passes through a particular point, you can use the point-slope form:

• Select the same slope as the original line, \( m = -\frac{5}{3} \)
• Use the coordinates of the given point.
• Apply these values in the point-slope equation \( y-y_1 = m(x-x_1) \).

This gives you a new line that runs in the same direction as the first line while passing through your chosen point.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which is 90 degrees. In terms of their slopes, two lines are perpendicular when the slopes are negative reciprocals of each other. If you have a line with a slope \( m \), a line perpendicular to it will have a slope \(-\frac{1}{m}\).

For illustration, consider a line with a slope of \(-\frac{5}{3}\). To find a perpendicular line's slope, you take the negative reciprocal, which results in \( \frac{3}{5} \). This is a key step in solving problems where you need to find the equation of a line perpendicular to a given line.

To write the equation of a perpendicular line passing through a specific point, follow these steps:

• Determine the slope of the new line, which must be the negative reciprocal of the original slope.
• Apply this slope and the given point's coordinates in the point-slope form \( y - y_1 = m(x - x_1) \).

These steps will guide you to construct the equation of a line that is perpendicular to the original and passes through your chosen point.

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to write the equation of a line. This form is especially handy because it directly shows the slope and the y-intercept of the line. The standard equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis.

When working with the equation of a line, converting it to slope-intercept form lets you quickly understand the line's direction and where it sits on the graph. For example, if you have a linear equation like \( 5x + 3y = 0 \), you can rearrange terms to solve for \( y \), giving you \( y = -\frac{5}{3}x \). From this new equation, it's clear that the slope \( m \) is \(-\frac{5}{3}\), and the y-intercept \( b \) is 0.

Being able to convert from different forms into slope-intercept form makes it easier to identify parallel and perpendicular lines and to graph them accurately. This knowledge is crucial whether you're handling linear equations in an algebra class or applying these concepts in geometry.