Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Passing through \((-2,4)\) and perpendicular to the line passing through \(\left(-5, \frac{1}{2}\right)\) and \(\left(-3, \frac{2}{3}\right)\)

Step-by-Step Solution

Answer

\( y = -12x - 20 \).

Step 1: Find slope of given line

Slope through \( (-5, 1/2) \) and \( (-3, 2/3) \):
\( m_1 = \frac{2/3 - 1/2}{-3 - (-5)} = \frac{4/6 - 3/6}{2} = \frac{1/6}{2} = \frac{1}{12} \).

Step 2: Find perpendicular slope

The perpendicular slope is \( m = -1/m_1 = -12 \).

Step 3: Write the equation

Using point-slope with \( (-2, 4) \):
\( y - 4 = -12(x - (-2)) = -12(x + 2) \)
\( y = -12x - 24 + 4 = -12x - 20 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a straight line. It is very convenient to use, especially when you're trying to graph a line quickly or determine key characteristics like slope and y-intercept. The standard equation in slope-intercept form is:

• \(y = mx + b\)

Here, \(m\) represents the slope of the line, which tells us how steep the line is, and \(b\) is the y-intercept, indicating where the line crosses the y-axis.
To write the equation of a line in slope-intercept form, you need the slope and one point on the line. With these, you plug the slope into \(m\) and use the point to solve for \(b\). Once both values are known, you can easily find the equation of the line. This form is typically preferred for visualizing lines graphically and quickly understanding their behavior.

Perpendicular Lines

In the context of geometry, understanding perpendicular lines is crucial. Two lines are considered perpendicular if they intersect at a right angle (90 degrees).
When finding the relationship between two lines in a coordinate plane, the product of their slopes is key. For two lines to be perpendicular, the product of their slopes must be -1.
For example, if you have a line with slope \(m_1\), then any line perpendicular to it will have a slope \(m_2\) such that:

• \(m_1 \times m_2 = -1\)

This concept is perfect for solving problems that ask you to find a line perpendicular to another, which is a common task in geometry and algebra exercises.
Once you've established the slope of the original line, the slope of a line perpendicular to it can be found by taking the negative reciprocal of the original slope. Understanding this relationship can help solve various geometric problems related to lines.

Finding Slope

Finding the slope of a line involves using a simple mathematic formula. The slope, often represented as \(m\), is calculated using the change in the y-coordinates divided by the change in the x-coordinates.
The formula to determine the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula tells us how much a line rises vertically (change in y) for every unit it moves horizontally (change in x).
For example, consider the points \((-5, \frac{1}{2})\) and \((-3, \frac{2}{3})\). By applying the formula, you calculate the slope as follows: replace the coordinates into the formula to get the slope \(m = \frac{\frac{2}{3} - \frac{1}{2}}{-3 - (-5)} = \frac{1}{12}\).
Understanding how to find the slope is a fundamental skill in algebra and is essential for studying the behavior of lines in mathematics.