Problem 54 - College Algebra with Corequisite Support

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

Step 1: Identify the slope of the given line

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

Step 2: Calculate the slope of the perpendicular line

Lines that are perpendicular to each other have slopes that are negative reciprocals. The slope of the given line is \(-2\), so the slope of the perpendicular line will be the negative reciprocal: \( m = \frac{1}{2} \).

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

The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \( m \) is the slope. Using the point \((-4, -1)\) and slope \(\frac{1}{2}\), the equation becomes: \( y + 1 = \frac{1}{2}(x + 4) \).

Step 4: Simplify the equation

Distribute the \(\frac{1}{2}\): \( y + 1 = \frac{1}{2}x + 2 \). Then, subtract 1 from both sides to solve for \( y \): \( y = \frac{1}{2}x + 1 \). This is the equation of the line perpendicular to the given line and passing through the point \((-4, -1)\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line. It's expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, which tells us how steep the line is, and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful because it allows you to quickly identify the slope and y-intercept just by looking at the equation.

Using an example equation, such as \( y = -2x + 4 \), tells us that:

• The slope \( m \) is \(-2\).
• The y-intercept \( b \) is 4. This means the line crosses the y-axis at \( (0, 4) \).

Having this format aids in graphing the line, as you can start at the y-intercept and use the slope to find other points on the line. For instance, from the point \((0, 4)\), you can move down 2 units and right 1 unit repeatedly to plot more points.

Point-Slope Form

The point-slope form is another method for writing the equation of a line, which is useful when you know a point on the line and the slope of the line. The general formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point, and \( m \) is the slope.

To use this form, let's take a point \((-4, -1)\) and a slope \( \frac{1}{2} \). Plugging these into the formula gives:

• Start with the formula: \( y - (-1) = \frac{1}{2}(x - (-4)) \).
• Simplify to get: \( y + 1 = \frac{1}{2}(x + 4) \).

This equation allows you to find the line that passes through the point \((-4, -1)\) with the specified slope. You can then simplify this equation into the slope-intercept form, making it easier to analyze or graph.

Negative Reciprocals

In geometry, two lines are perpendicular if they intersect at a right angle. A key concept here is the relationship between their slopes. Specifically, the slopes of perpendicular lines are negative reciprocals. This means if one line has a slope of \( m \), then the other line will have a slope of \( -\frac{1}{m} \).

For example, consider a line with a slope of \(-2\). The negative reciprocal of \(-2\) is \( \frac{1}{2} \). This tells us that any line with a slope of \( \frac{1}{2} \) would be perpendicular to the original line with slope \(-2\).

Knowing about negative reciprocals helps in quickly determining the slope needed to find a perpendicular line, a common requirement in many geometry problems. Understanding this relationship is essential for writing equations of lines that need to be perpendicular to a given line.