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Chapter 1: Problem 32

Find an equation of the line that satisfies the given conditions. Through \(\left(\frac{1}{2},-\frac{2}{3}\right) ; \quad\) perpendicular to the line \(4 x-8 y=1\)

Short Answer

Expert verified
The equation of the line is \(y = -2x + \frac{1}{3}\).

Step by step solution

01

Identify the slope of the given line

The given line is in the form of the equation \(4x - 8y = 1\). First, we need to rewrite it in the slope-intercept form \(y = mx + b\) to find its slope. By isolating \(y\), we get: \[4x - 8y = 1\]\[-8y = -4x + 1\]\[y = \frac{1}{2}x - \frac{1}{8}\]The slope \(m\) of the given line is \(\frac{1}{2}\).
02

Find the perpendicular slope

Lines that are perpendicular have slopes that are negative reciprocals of each other. The slope of the given line is \(\frac{1}{2}\). To find the negative reciprocal, flip the fraction and change the sign, resulting in a perpendicular slope of \(-2\).
03

Use the point-slope form to find the equation

We have the perpendicular slope \(-2\) and the point \(\left(\frac{1}{2}, -\frac{2}{3}\right)\). Apply the point-slope form of the equation of a line: \[y - y_1 = m(x - x_1)\]Substituting \(m = -2\) and the point, we get:\[y + \frac{2}{3} = -2\left(x - \frac{1}{2}\right)\]
04

Simplify the equation

Expand and simplify the equation:\[y + \frac{2}{3} = -2x + 1\]Subtract \(\frac{2}{3}\) from both sides:\[y = -2x + 1 - \frac{2}{3}\]To combine the constants, convert \(1\) to \(\frac{3}{3}\):\[y = -2x + \frac{3}{3} - \frac{2}{3}\]Thus, \[y = -2x + \frac{1}{3}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a line so that we can easily identify its slope and y-intercept. The general formula is given by \[y = mx + b\] where:
  • \(m\) is the slope of the line, indicating how steep it is.
  • \(b\) is the y-intercept, showing where the line crosses the y-axis.
To convert a standard form equation, like \(4x - 8y = 1\), into slope-intercept form:
  • Isolate \(y\) by solving the equation \(-8y = -4x + 1\).
  • Divide every term by -8 to give \(y\) a coefficient of 1.
  • You get \(y = \frac{1}{2}x - \frac{1}{8}\).
Now, it's easy to see that the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is \(-\frac{1}{8}\). This form helps to quickly identify parallel and perpendicular lines by comparing slopes.
Negative Reciprocal
Perpendicular lines have a special relationship when it comes to their slopes: the slopes are negative reciprocals of one another. If you have a line with a slope of \(m\), the slope \(m_\perp\) of a line perpendicular to it will be \(-\frac{1}{m}\). Here鈥檚 how it works:
  • Start with the slope of the original line, which is \(\frac{1}{2}\).
  • Flip the fraction \(\frac{1}{2}\) to get \(2\).
  • Then, change the sign to make it \(-2\).
So, the slope of a line perpendicular to one with a slope of \(\frac{1}{2}\) is \(-2\). This concept is key when working with lines that intersect at right angles, as it simplifies finding the slope of a perpendicular line.
Point-Slope Form
The point-slope form is useful when you want to write an equation of a line, especially if you have a point on the line and the slope. The formula is:\[y - y_1 = m(x - x_1)\]
  • \((x_1, y_1)\) is a specific point on the line.
  • \(m\) is the slope of the line.
Let's use this form to find the equation of a line passing through a given point, say \(\left(\frac{1}{2}, -\frac{2}{3}\right)\), with a slope of \(-2\):
  • Plug \(m = -2\), \(x_1 = \frac{1}{2}\), and \(y_1 = -\frac{2}{3}\) into the equation.
  • Your formula becomes: \[y + \frac{2}{3} = -2\left(x - \frac{1}{2}\right)\]
After substituting, expand and simplify it to match a linear equation format, maintaining its original components. This helps us express lines clearly and concisely.

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