Problem 20 - College Algebra

Question

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

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 1 = -x - 4\), and in slope-intercept form is \(y = -x - 5\)

Step 1: Find Point-Slope Form

Substitute the slope, -1, and the point, (-4, -1), into the point-slope form equation \(y - y_1 = m(x - x_1)\). This gives \(y - (-1) = -1(x - (-4))\). Simplify to get \(y + 1 = -1(x + 4)\). This can be further simplified to \(y + 1 = -x - 4\) which is the equation in point-slope form.

Step 2: Convert to Slope-Intercept Form

To get the slope-intercept form, rearrange the point-slope equation \(y + 1 = -x - 4\) to make y the subject and simplify it. This gives \(y = -x - 4 - 1\). Simplifying this further you get \(y = -x - 5\), which is the equation in slope-intercept form.

Key Concepts

Point-Slope Form

Point-slope form is a way to describe the equation of a line using its slope and a single point that lies on the line. The general formula for point-slope form is:

• \( y - y_1 = m(x - x_1) \)

You need two pieces of information to use this format: the slope \( m \) of the line and one known point \((x_1, y_1)\) on the line.
This form is very handy because it directly ties the linear equation to the geometric elements of a line – its steepness and location. For instance, given a slope of -1 and a point \((-4, -1)\), you plug these values into the formula like this:
\( y - (-1) = -1(x - (-4)) \).
With a bit of simplification, such as replacing double negatives, this becomes:
\( y + 1 = -1(x + 4) \).
So, now you have the equation in point-slope form, clearly connecting the equation to the graph's features.

Slope-Intercept Form

Slope-intercept form is another way to express a linear equation, and it is popular for its simplicity.

• The formula is: \( y = mx + b \)

Here, \( m \) represents the slope of the line, similar to the point-slope form. The letter \( b \) stands for the y-intercept, which is the point where the line crosses the y-axis.
Slope-intercept form is useful because it tells you at a glance where the line intersects the y-axis and how steep it is. To convert a point-slope form equation to slope-intercept form, you need to rearrange and solve for \( y \).
Starting with \( y + 1 = -x - 4 \) from the point-slope form, isolate \( y \) to get:
\( y = -x - 4 - 1 \) then simplify to obtain:
\( y = -x - 5 \).
This equation now neatly provides the slope \(-1\) and y-intercept \(-5\), making it straightforward to graph the line.

Slope of a Line

The slope of a line indicates how steep the line is, showing the rate of change between the y-coordinate and the x-coordinate on a graph.

• It is calculated as \( \frac{\text{rise}}{\text{run}} \)

In algebra, the slope is represented by the letter \( m \) and can be calculated if two points are known. The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Understanding the slope is crucial because it helps predict how the line behaves as it moves across the graph. A positive slope means the line ascends from left to right, whereas a negative slope, like \(-1\) in our example, means it descends instead.
This consistent rate of descent is a key aspect when solving and graphing linear equations both in point-slope and slope-intercept forms.