Problem 32 - Algebra and Trigonometry

Question

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 \(4 x-8 y=1\)

Step-by-Step Solution

Answer

The equation is \(y = -2x + \frac{1}{3}\).

Step 1: Find the slope of the given line

The equation of the given line is in the form of \(4x - 8y = 1\). We need to rewrite it in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. Start by solving for \(y\):\[4x - 8y = 1\] \[\Rightarrow -8y = -4x + 1\] \[\Rightarrow y = \frac{1}{2}x - \frac{1}{8}\]. The slope of the line is \(m = \frac{1}{2}\).

Step 2: Determine the slope of the perpendicular line

A line perpendicular to another has a slope that is the negative reciprocal of the original line's slope. The original slope is \(\frac{1}{2}\), so the negative reciprocal is \(m = -2\).

Step 3: Use point-slope form to find the equation

We use the point \(\left(\frac{1}{2}, -\frac{2}{3}\right)\) and the slope \(-2\) to write the equation in point-slope form: \[y - y_1 = m(x - x_1)\] where \(x_1 = \frac{1}{2}\) and \(y_1 = -\frac{2}{3}\). Substituting these values, we get: \[y + \frac{2}{3} = -2\left(x - \frac{1}{2}\right)\].

Step 4: Simplify the equation

Now we simplify the equation: \[y + \frac{2}{3} = -2x + 1\]. Subtract \(\frac{2}{3}\) from both sides: \[y = -2x + 1 - \frac{2}{3}\]. Convert \(1\) to a fraction: \(1 = \frac{3}{3}\), \[y = -2x + \frac{3}{3} - \frac{2}{3}\], \[y = -2x + \frac{1}{3}\].

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a very useful way to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept, or the point where the line crosses the y-axis.

To convert an equation to the slope-intercept form, you need to solve for \( y \) in terms of \( x \). For example, if you begin with \( 4x - 8y = 1 \), you would aim to isolate \( y \) on one side. Here’s a step-by-step process:

• First, move terms involving \( x \) to the other side of the equation: \( -8y = -4x + 1 \).
• Then, solve for \( y \) by dividing every term by \(-8\), resulting in \( y = \frac{1}{2} x - \frac{1}{8} \).

Now, you can easily identify the slope \( m = \frac{1}{2} \). This form makes it straightforward to graph lines, and find how one variable changes with another.

Negative Reciprocal

When two lines are perpendicular to each other, their slopes are negative reciprocals. What does this mean? It means that you take the reciprocal of the first line's slope and then change the sign.

If the slope of the first line is \( \frac{a}{b} \), the negative reciprocal is \( -\frac{b}{a} \). This relationship is key when determining the slope of a line perpendicular to another.

• In our problem, the original line's slope was \( \frac{1}{2} \).
• The negative reciprocal is \( -2 \), since flipping \( \frac{1}{2} \) gives \( 2 \), and changing the sign becomes \( -2 \).

Using this relationship, we can ensure that the lines meet at a right angle, an important feature for many geometric applications.

Point-Slope Form

The point-slope form of an equation is very handy when you know the slope of a line and a point through which it passes. It is formulated as \( y - y_1 = m(x - x_1) \).

This form is especially useful when you need to derive an equation from given conditions. Here's how you can utilize it:

• Choose a point on the line, like \( (x_1, y_1) \) which is \( \left(\frac{1}{2}, -\frac{2}{3}\right) \) in our case.
• Use the known slope \( m = -2 \).

Plugging in these values into the formula gives

• \( y + \frac{2}{3} = -2(x - \frac{1}{2}) \)
• This equation can be simplified further to obtain the final line equation in either slope-intercept or another preferred form.

Point-slope form is a powerful intermediary step to express lines clearly and usefully.