Problem 30 - Thomas Calculus

Question

In Exercises 17–30, write an equation for each line described. Passes through \((0,1)\) and is perpendicular to the line \(8 x-13 y=13\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{13}{8}x + 1\).

Step 1: Understand the Problem

We need to find the equation of a line that passes through the point \((0,1)\) and is perpendicular to the line with the equation \(8x - 13y = 13\).

Step 2: Find the Slope of the Given Line

Convert the given line equation to the slope-intercept form \(y = mx + b\). Start by isolating \(y\):\[8x - 13y = 13\]\[13y = 8x - 13\]\[y = \frac{8}{13}x - \frac{13}{13}\]The slope \(m\) of this line is \(\frac{8}{13}\).

Step 3: Find the Perpendicular Slope

The slope of a line perpendicular to another is the negative reciprocal of the other line's slope. So, the perpendicular slope \(m'\) is \[m' = -\frac{13}{8}\].

Step 4: Write the Equation of the Perpendicular Line

The perpendicular line passes through \((0,1)\) and has the slope \(-\frac{13}{8}\). Use the point-slope form \(y - y_1 = m(x - x_1)\) with \((x_1, y_1) = (0, 1)\):\[y - 1 = -\frac{13}{8}(x - 0)\]\[y - 1 = -\frac{13}{8}x\]\[y = -\frac{13}{8}x + 1\].

Step 5: Simplify the Equation if Necessary

The slope-intercept form of the perpendicular line through \((0,1)\) is already in the simplest form:\[y = -\frac{13}{8}x + 1\].

Key Concepts

Slope-Intercept Form

The slope-intercept form is a simple way of expressing the equation of a line. The format is \(y = mx + b\), where

• \(m\) represents the slope of the line,
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

Using this form, it's easy to analyze and graph lines. For instance, the slope tells us how steep the line is and in which direction it tilts. A positive slope means that the line rises as it moves from left to right, while a negative slope has it falling.
Applying this to our exercise, when the given line's equation is manipulated into the slope-intercept form, it helps quickly identify the slope for further calculations, assisting in determining the perpendicular slope.

Point-Slope Form

The point-slope form is used for writing the equation of a line when you know the slope and a single point on the line. The format is \(y - y_1 = m(x - x_1)\), where

• \((x_1, y_1)\) represents the coordinates of a point on the line,
• \(m\) is the slope.

This form is particularly useful when you have these two pieces of information, as is the case with the perpendicular line we want to find.
For example, our problem gives us a point \((0,1)\) and together with the perpendicular slope, we use the point-slope form to easily derive the line equation. It's like filling in the blanks of a pattern once you know the key details.

Negative Reciprocal

A negative reciprocal is key when determining perpendicular lines. If a line has slope \(m\), then a line perpendicular to it will have the slope \(-\frac{1}{m}\). This happens because the product of the slopes of two perpendicular lines is \(-1\).
This concept can be a little tricky but it's important for problems involving perpendicular lines. For instance, in our exercise, the slope of the given line is \(\frac{8}{13}\). To find the slope of a line perpendicular to this, simply take the negative reciprocal, leading you to \(-\frac{13}{8}\). This new slope then guides the writing of the equation for the perpendicular line.

Equation of a Line

The equation of a line is a mathematical way of representing a line in the x-y plane. There are various forms like the slope-intercept and point-slope forms, each useful in different scenarios.
The slope-intercept form\(y = mx + b\) provides immediate insights on the slope and y-intercept, making it handy for graphing. On the other hand, the point-slope form\(y - y_1 = m(x - x_1)\) is useful when you know a point on the line and the slope.
In the exercise, we've used these forms to first determine the slope of the original line and then find its perpendicular. After identifying the slope and a point, we've expressed the new line using these forms, showcasing the equation \(y = -\frac{13}{8}x + 1\) in the slope-intercept form, clearly representing the perpendicular line.