Problem 55 - College Algebra

Question

For the following exercises, write an equation for the line described. Write an equation for a line perpendicular to \(p(t)=3 t+4\) and passing through the point (3,1).

Step-by-Step Solution

Answer

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

Step 1: Determine the Slope of Line p(t)

The given equation for the line is \( p(t) = 3t + 4 \). This is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. Thus, the slope of the given line, \( p(t) \), is 3.

Step 2: Find the Negative Reciprocal for Perpendicular Slope

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, take the negative reciprocal of 3: \[ m = -\frac{1}{3} \] This will be the slope of our new line.

Step 3: Use the Point-Slope Form to Write Line Equation

We need to write the equation using the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \),where \( (x_1, y_1) \) is the point (3,1), and \( m \) is \(-\frac{1}{3}\). Substitute these values into the equation:\[ y - 1 = -\frac{1}{3}(x - 3) \]

Step 4: Simplify the Equation to Slope-Intercept Form

Expand the equation from point-slope form and simplify to slope-intercept form:\[ y - 1 = -\frac{1}{3} x + 1 \]Add 1 to both sides to solve for \( y \): \[ y = -\frac{1}{3} x + 2 \]This equation represents the line perpendicular to \( p(t) \) and passing through the point (3,1).

Key Concepts

Slope-Intercept Form

The slope-intercept form of an equation of a line is one of the most straightforward ways to express linear equations. The format of a slope-intercept equation is \( y = mx + b \), where:

• \( m \) represents the slope of the line
• \( b \) denotes the y-intercept, which is where the line crosses the y-axis

This form is popular because it provides immediate insights about the line. The slope \( m \) tells you how steep the line is and whether it rises or falls as it moves from left to right. A positive slope means the line ascends, while a negative slope means it descends. The y-intercept \( b \) serves as the starting point of the line on the graph.
For instance, the equation \( p(t) = 3t + 4 \) is in slope-intercept form with a slope of 3 and a y-intercept of 4. This indicates a relatively steep line that crosses the y-axis at the point (0,4).

Point-Slope Form

When a particular point on a line and its slope are known, the point-slope form comes into use. The formula for this form is \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a specific point on the line
• \( m \) is the slope of the line

This format is particularly valuable when you don't have the y-intercept but do know a point through which the line passes. Using the point-slope form helps to quickly generate an equation for the line.
In the context of the exercise, we used the point-slope form to write the equation of a line passing through the point \((3,1)\) with a slope of \(-\frac{1}{3}\). By substituting these into the formula, we obtain: \( y - 1 = -\frac{1}{3}(x - 3) \). This equation highlights the relationship between the slope and a given point on the new perpendicular line.

Negative Reciprocal

The concept of negative reciprocal is crucial when dealing with perpendicular lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. This means that the product of the slopes of the two lines must equal \(-1\).
To find the negative reciprocal, follow these steps:

• Take the reciprocal of the original slope (flip the fraction)
• Change its sign from positive to negative or vice versa

So, if the original slope is 3, its negative reciprocal is \(-\frac{1}{3}\), as calculated by flipping 3 to \(\frac{1}{3}\) and then changing the sign to negative. This new slope \(-\frac{1}{3}\) was used in the exercise to establish the line that is perpendicular to the given line \(p(t) = 3t + 4\). The concept ensures precise orientation of the line, forming a perfect 90-degree angle with the original.