Problem 54 - College Algebra

Question

Write an equation for a line perpendicular to \(h(t)=-2 t+4\) and passing through the point \((-4,-1)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Line

The given line is represented by the equation \( h(t) = -2t + 4 \). This is in slope-intercept form \( y = mx + b \), where \( m = -2 \) and \( b = 4 \). Here, the slope of the line is \( -2 \).

Step 2: Find the Perpendicular Slope

To find the perpendicular slope, take the negative reciprocal of the original slope \(-2\). The negative reciprocal of \(-2\) is \(\frac{1}{2}\), so the slope \( m_{\perp} = \frac{1}{2} \).

Step 3: Use the Point-Slope Formula

The point-slope form of the equation of a line is given by \( y - y_1 = m(x - x_1) \). Here, \( m = \frac{1}{2} \) (the perpendicular slope), and \( (x_1, y_1) = (-4, -1) \). Substitute these values into the formula to get: \[ y + 1 = \frac{1}{2}(x + 4) \].

Step 4: Convert to Slope-Intercept Form

To convert the equation from point-slope to slope-intercept form, solve for \( y \). Start by distributing the slope: \[ y + 1 = \frac{1}{2}x + \frac{1}{2} \times 4 \]. This simplifies to \( y + 1 = \frac{1}{2}x + 2 \). Now, subtract 1 from both sides to find \( y \): \( y = \frac{1}{2}x + 1 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that its slope and y-intercept are immediately apparent. This form is given by the equation \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
For example, in the line equation \( h(t) = -2t + 4 \), it can be rewritten as \( y = -2x + 4 \) by replacing \( t \) with \( x \). Here, the slope \( m \) is \(-2\), indicating how steep the line is.
Using the slope-intercept form is particularly helpful when plotting the line on a graph because you can start plotting at the y-intercept \((0, b)\) and use the slope \( m \) to determine the rise over run to plot the next point.

Point-Slope Formula

The point-slope formula is used to find the equation of a line when you know the slope and a single point on the line. This formula is written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a known point on the line.
This approach is particularly useful when dealing with lines that are not easily expressed with a y-intercept, or when you need to find an equation for a new line, such as one that is perpendicular to another. If we have a perpendicular slope \( \rac{1}{2}\) and a point \((-4, -1)\), substituting into the formula gives \( y + 1 = \rac{1}{2}(x + 4) \).
By plugging the point and the slope into the point-slope formula, you can establish the line's equation efficiently. This equation can also be rearranged to slope-intercept form, making it easier to graph.

Negative Reciprocal

The concept of a negative reciprocal is crucial when dealing with perpendicular lines. If two lines are perpendicular to each other, the slopes of these lines have a unique relationship: they are negative reciprocals of each other.
To find the negative reciprocal of a slope, take its reciprocal (flip the fraction) and change its sign. For example, if the original slope is \(-2\), its reciprocal is \( \rac{1}{-2} = -\frac{1}{2} \). Changing its sign gives the negative reciprocal, which is \( \rac{1}{2} \).
Practically, this means that if you have a line with a slope \( m \), the slope of a line perpendicular to it will be \(-\frac{1}{m} \). This relationship allows you to easily find the slope when solving problems involving perpendicular lines, such as finding the equation of a line that intersects another at a 90-degree angle.