Problem 33 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,-1)\) and \((4,-1)\)

Step-by-Step Solution

Answer

The equations of the line in point-slope and slope-intercept forms are both \(y = -1\)

Step 1: Calculate the Slope

First task is to calculate the slope (m) of the line using the formula \((y_2 - y_1) / (x_2 - x_1)\). The points are (-3, -1) where \((x_1, y_1) = (-3, -1)\) and (4, -1) where \((x_2, y_2) = (4, -1)\). So, \(m = (-1 - (-1)) / (4 - (-3)) = 0\). This tells us that the line is horizontal.

Step 2: Find the Point-Slope Form

Next, we substitute the values for the slope (m), and one of the points \((-3, -1)\) into the point-slope form of equation \(y - y_1 = m(x - x_1)\). On substitution, we get \(y - (-1) = 0 * (x - (-3))\). Cleaning this equation up, we obtain the equation \(y = -1\) which is the final point-slope form equation.

Step 3: Find the Slope-Intercept Form

In order to convert the equation to slope-intercept form (\(y = mx + b\)) where 'b' is the y-intercept, looking at the line equation \(y = -1), we see that the slope m = 0, and y-intercept b = -1. So the final slope-intercept form equation will be \(y = 0x - 1\), which simplifies to \(y = -1\). In this case, the slope-intercept form is the same as the point-slope form because the line is horizontal.

Key Concepts

Point-Slope Form

The point-slope form is a way to describe a line on the coordinate plane using one point on the line and the slope of the line. This form is especially useful when you know a point the line passes through, as well as the slope. The general equation for the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here:

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

When you substitute the point

• \((-3, -1) \)

and the slope

• \( m = 0 \)

into the equation, you simplify it to \( y = -1 \). This indicates a horizontal line passing through \( y = -1 \).
One of the advantages of point-slope form is its ability to describe the line with just a single point and a slope, making it a quick way to find or describe a line.

Slope-Intercept Form

Slope-intercept form is one of the most popular forms for writing equations of lines. It's incredibly useful for easily understanding a line's behavior. The general formula is given by:

• \( y = mx + b \)

Where:

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

In this particular example, since the line equation is \( y = -1 \), it can also be represented as \( y = 0x - 1 \), showing clearly:
The slope \( m = 0 \)
The y-intercept \( b = -1 \)
Since \( m = 0 \), this tells us the line is horizontal. In horizontal lines, each point on the line shares the same y-coordinate. Therefore, regardless of the value of \( x \), \( y \) will always equal -1.

Slope Calculation

Calculating the slope of a line is a fundamental skill in algebra that helps us understand how steep a line is. The slope is determined by the "rise" over the "run" between two points on the line. The formula to calculate the slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two different points on the line.

In our specific problem, we used the points \((-3, -1)\) and \((4, -1)\).
By plugging in these points, the formula becomes:

• \( m = \frac{-1 - (-1)}{4 - (-3)} = \frac{0}{7} = 0 \)

This means the line has a slope of 0, which characterizes a horizontal line. A horizontal line has no rise—it stays flat.