Problem 54 - College Algebra

Question

For the following exercises, write an equation for the line described. 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 of the line is \( y = \frac{1}{2} x + 1 \).

Step 1: Identify the slope of the given line

The given line is in the form of a linear equation \( h(t) = -2t + 4 \). Here, the slope \( m \) of the line is -2.

Step 2: Determine the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of one another. So, to find the slope of the new line, take the negative reciprocal of -2, which is \( \frac{1}{2} \).

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

Use the point-slope form of a linear equation: \( y - y_1 = m(x - x_1) \). Substitute \( m = \frac{1}{2} \), \( x_1 = -4 \), and \( y_1 = -1 \) into the equation: \( y + 1 = \frac{1}{2}(x + 4) \).

Step 4: Simplify to slope-intercept form

Distribute \( \frac{1}{2} \) in the equation: \( y + 1 = \frac{1}{2}x + 2 \). Then, isolate \( y \) by subtracting 1 from both sides: \( y = \frac{1}{2}x + 1 \).

Key Concepts

Slope of a line

The slope of a line is a key concept in coordinate geometry that represents the direction and steepness of a line. The slope is often symbolized by the letter \( m \). It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, this is expressed as:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• A zero slope indicates a horizontal line, while an undefined slope indicates a vertical line.

In our example, the line equation is given as \( h(t) = -2t + 4 \), where the slope \( m \) is \(-2\). Identifying the slope is the first crucial step in solving problems involving equations of lines, especially when you're dealing with perpendicular lines.

Point-slope form

The point-slope form is a useful way to write the equation of a line when you know the slope and one point on the line. This form is particularly handy because it directly uses the slope and a point to generate the equation of the line. The point-slope form is written as:\[y - y_1 = m(x - x_1)\]Here:

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

For instance, in the exercise, after determining the perpendicular slope as \( \frac{1}{2} \), and with the point \((-4, -1)\), the equation is set up as \( y + 1 = \frac{1}{2}(x + 4) \). This form makes it straightforward to later rearrange the equation into different forms, like slope-intercept form.

Slope-intercept form

The slope-intercept form is one of the most commonly used forms of linear equations, thanks to its simplicity and directness. It is generally written as:\[y = mx + b\]Here:

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

This form is useful because it quickly reveals both the slope and the y-intercept of the line, making graphs easier to plot. In the exercise given, after using the point-slope form, the equation is simplified to the slope-intercept form as \( y = \frac{1}{2}x + 1 \). By simplifying the equation to this form, it becomes easier to identify how the line will behave on a graph, as you can clearly see the starting point at the y-axis and the angle at which the line inclines.