Problem 11 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (4,-7) and perpendicular to the line whose equation is \(x-2 y-3=0\)

Step-by-Step Solution

Answer

The equation of the line passing through (4,-7) and perpendicular to the line whose equation is \(x-2y-3=0\) is \(y+7=-2(x-4)\) in point-slope form and \(-2x+y+15=0\) in general form.

Step 1: Find the slope of given line

First, rewrite the equation \(x-2y-3=0\) into slope-intercept form by solving for y to find the slope. In this form, the equation will be like \(y=mx+b\) where m is the slope. It gives us \(y=\frac{1}{2}x-\frac{3}{2}\). Thus, the slope of given line is \(\frac{1}{2}\).

Step 2: Find the slope of the perpendicular line

Since a line is perpendicular to another if the slopes of the two lines are negative reciprocals of each other, the slope of the line that is perpendicular to the given line will be the negative reciprocal of \(\frac{1}{2}\). So, the slope of the required line is -2.

Step 3: Write the equation in point-slope form

Use the point-slope form \(y-y_1=m(x-x_1)\), where \((x_1,y_1)\) is a point on the line and m is the slope. Substitute m=-2 and \((x_1,y_1)\) with (4,-7) to get the equation of the line. Therefore, the equation of the line in point-slope form is \(y+7=-2(x-4)\).

Step 4: Write the equation in general form

Expand the equation from step 3 and then group the terms of x and y on one side to get the equation in the general form. The equation becomes \(-2x+y+15=0\).

Key Concepts

Point-Slope Form

The point-slope form is incredibly useful when we have a single point and the slope of a line. It's represented by the equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line and \(x_1, y_1\) are the coordinates of the given point.

To use this form, simply replace \(m\) with the line's slope and \(x_1, y_1\) with the coordinates of the known point. It's the best starting point for writing an equation when these pieces of information are available, because it directly incorporates both the slope and a specific point to define the line completely.

Perpendicular Lines

Understanding perpendicular lines is essential in geometry. Two lines that intersect to form right angles are called perpendicular. Here's the interesting part: the slopes of two perpendicular lines are negative reciprocals of each other. This means if one line has a slope of \(a\), the other will have a slope of \(\frac{-1}{a}\).

This relationship is handy when you need to find the equation of a line perpendicular to a given line - just take the negative reciprocal of the original line's slope! Remember, for horizontal and vertical lines, things are a bit different; a vertical line has an undefined slope, and a horizontal line has a slope of 0.

Slope-Intercept Form

The slope-intercept form \(y = mx + b\) is another popular way to represent a line. In this equation, \(m\) is the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This form is incredibly user-friendly because it gives you the rise-over-run ratio directly and shows at which point the line will hit the y-axis.

Converting to slope-intercept form often involves solving for \(y\) and is a great first step in graphing the equation or finding the perpendicular slope, as the slope \(m\) can be easily identified.

Negative Reciprocal Slope

The negative reciprocal slope is a critical concept when dealing with perpendicular lines. If a line has a slope of \(m\), its perpendicular counterpart will have a slope of \(\frac{-1}{m}\). This means if you have a line with a positive slope, the perpendicular line's slope will be negative, and vice versa.

In the realm of slopes, 'reciprocal' means you flip the fraction (if the slope isn't a fraction, simply put it over 1). Don't forget to change the sign to the opposite to get the negative reciprocal. For instance, if a line has a slope of 3 (which is the same as \(\frac{3}{1}\)), a line perpendicular to it would have a slope of \(\frac{-1}{3}\). This concept is foundational when determining the orientation of lines to each other in a coordinate plane.