Problem 45 - A Graphical Approach to College Algebra

Question

Find the equation of the line satisfying the given conditions, giving it in slope-intercept form if possible. Through \((1,6),\) perpendicular to \(3 x+5 y=1\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{5}{3}x + \frac{13}{3}\).

Step 1: Identify the slope of the given line

First, we need to identify the slope of the given line, which is written in the form \(3x + 5y = 1\). To do this, we'll convert it to the slope-intercept form \(y = mx + b\). To solve for \(y\), subtract \(3x\) from both sides and then divide by \(5\) to get: \(y = -\frac{3}{5}x + \frac{1}{5}\). The slope \(m\) of the given line is \(-\frac{3}{5}\).

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, we find the negative reciprocal of \(-\frac{3}{5}\), which is \(\frac{5}{3}\). So, the slope of the line we seek is \(\frac{5}{3}\).

Step 3: Use point-slope form

We have a point \((1, 6)\) and a slope \(\frac{5}{3}\). To find the equation of the line, we use the point-slope form formula: \(y - y_1 = m(x - x_1)\). Substituting the known values, \(y - 6 = \frac{5}{3}(x - 1)\).

Step 4: Convert to slope-intercept form

Now we convert \(y - 6 = \frac{5}{3}(x - 1)\) into the slope-intercept form \(y = mx + b\). First, distribute the slope: \(y - 6 = \frac{5}{3}x - \frac{5}{3}\). Then, add 6 to both sides to solve for \(y\): \(y = \frac{5}{3}x + \frac{13}{3}\). The slope-intercept form of the equation is \(y = \frac{5}{3}x + \frac{13}{3}\).

Key Concepts

Slope-Intercept Form

The Slope-Intercept Form is a powerful and very common way to express the equation of a line. It follows the structure \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is advantageous because it gives clear insight into the line's steepness and where it crosses the y-axis.

To convert an equation into this form, start by isolating \(y\). For instance, if you have the equation \(3x + 5y = 1\), subtract \(3x\) from both sides to yield \(5y = -3x + 1\). Then, divide each term by 5 to isolate \(y\): \(y = -\frac{3}{5}x + \frac{1}{5}\). Now, the slope \(m\) is \(-\frac{3}{5}\) and the y-intercept \(b\) is \(\frac{1}{5}\).

This format makes it easy to graph or understand the line. You can instantly see how steep the line is, and where exactly it meets the y-axis.

Point-Slope Form

The Point-Slope Form is hugely beneficial when you know a specific point on a line and the line's slope. The equation is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) represent a point on the line.

For example, if you are given the point \((1,6)\) and the slope \(\frac{5}{3}\), you can substitute these values into the formula: \(y - 6 = \frac{5}{3}(x - 1)\). This form is especially useful because it highlights the relationship between the slope and a specific point on the line, making it a simple way to create the equation of a line.

When you convert this to Slope-Intercept Form, you get another common and graph-friendly version of the equation, which can be straightforwardly carried out by simplifying the Point-Slope Form equation.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. For two lines to be perpendicular, the product of their slopes must equal \(-1\). In simpler terms, one line's slope is the negative reciprocal of the other line's slope.

For example, if you have a line with a slope of \(-\frac{3}{5}\), the slope of a line that is perpendicular to it will be \(\frac{5}{3}\).

• Calculate the negative reciprocal by flipping the fraction and changing the sign.
• This relationship helps in finding the equation of a line perpendicular to a given line.

Understanding the concept of perpendicular slopes allows you to efficiently solve problems requiring the creation of perpendicular lines, such as generating the equation of a line that passes through a set point and intersects another line at a right angle.