Problem 19 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-3,-1)\) and \((2,4)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = x + 2\) and in slope-intercept form is also \(y = x + 2\)

Step 1: Calculate the slope

The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[\(m = \frac{y_2 - y_1}{x_2 - x_1}\]\. So for the points (-3,-1) and (2, 4), the slope is calculated as: \[m = \frac{4 - (-1)}{2 - (-3)} = 1\]

Step 2: Write the point-slope form

The point-slope formula is \[y - y_1 = m(x - x_1)\]. With the calculated slope and one of the points, the equation of the line in point-slope form is: \[y - (-1) = 1 * (x - (-3)) \rightarrow y + 1 = 1 * (x + 3) \rightarrow y = x + 3 - 1 \], which simplifies to \[y = x + 2\]

Step 3: Write the slope-intercept form

The slope-intercept form \((y = mx + c)\) is obtained by rearranging the point-slope equation we just found in Step 2. However, as the equation is already in the form \[y = mx + c\] with \(m = 1\) and \(c = 2\), no further work is needed.

Key Concepts

Slope Calculation

Calculating the slope of a line is an essential skill in understanding the geometry of lines. The slope (\(m\) describes the steepness of a line. You can think of it as the "tilt" of the line that tells you how much the line rises or falls for every unit it moves horizontally.

To calculate the slope between two points, use the formula:

• Identify the coordinates of the two points, usually denoted as \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the \(y\) values \((y_2 - y_1)\) to find the rise, and subtract the \(x\) values \((x_2 - x_1)\) to find the run.
• Divide the rise by the run using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Let's apply this to the points (-3, -1) and (2, 4): We find the rise to be \(4 - (-1) = 5\) and the run to be \(2 - (-3) = 5\). Thus, \(m = \frac{5}{5} = 1\).

This tells us that the line rises 1 unit for every 1 unit it moves to the right, which is crucial for developing the equation of the line.

Equation of a Line

Once you know the slope and a point on the line, you can develop the equation of the line. The point-slope form comes in handy for this purpose. It uses a known point on the line and the slope to create an equation. The point-slope formula is:

• \(y - y_1 = m(x - x_1)\)
• \((x_1, y_1)\) is a point on the line.
• \(m\) is the slope you have calculated.

By inserting the slope \(m = 1\) and using the point (-3, -1), we arrive at the point-slope equation: \(y + 1 = 1(x + 3)\).

This equation simplifies to \(y = x + 2\) as shown, which gives us a simple linear relationship between \(x\) and \(y\), effectively describing the line's path.

The beauty of this method is that it transforms the abstract slope and a point into an easily useable mathematical expression.

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a line, especially because it directly tells us both the slope and the y-intercept.

Its formula is \(y = mx + c\), where:

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

If you've followed along using the point-slope form and simplified correctly, you naturally arrive at the slope-intercept form. In our example, the equation simplification leads us to \(y = x + 2\).

Here, \(m = 1\) indicates every step to the right raises the line 1 unit, and \(c = 2\) shows the line crosses the y-axis at (0, 2).

The slope-intercept form is especially advantageous when graphing a line, as it gives immediate visual information about the starting point and the direction of the line's path.