Problem 32 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,6)\) and \((3,-2)\)

Step-by-Step Solution

Answer

The point-slope form of the equation is \(y - 6 = -4/3(x + 3)\), and the slope-intercept form is \(y = -4/3x + 2\).

Step 1: Calculate the Slope

Firstly, calculate the slope using the two given points \((-3,6)\) and \((3,-2)\). The slope (m) is given by the formula \(m = (y2 - y1) / (x2 - x1)\), so substituting the given coordinates into the formula gives us \(m = (-2 - 6)/(3 + 3) = -8/6 = -4/3\).

Step 2: Write the equation in point-slope form

The point-slope form equation is given by \(y - y1 = m(x - x1)\). Taking the given point \((-3,6)\) and the previously calculated slope, we get the point-slope form of the equation as \(y - 6 = -4/3(x + 3)\).

Step 3: Write the equation in slope-intercept form

The slope-intercept form is given as \(y = mx + b\). To find 'b', we can substitute x, y, and m into our equation resulting in \(6 = -4/3*(-3) + b\), which gives \(b = 6 - 4 = 2\). Hence the slope-intercept form of the line is \(y = -4/3x + 2\).

Key Concepts

Point-Slope Form

The point-slope form of a line’s equation is a powerful tool, especially when you have a point and the slope of that line. It's written as:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) represents a known point on the line, and \( m \) is the slope of the line. This format is ideal for quickly forming an equation when you have one point and the slope.
For instance, with the point \((-3, 6)\) and a slope of \(-4/3\), we plug these into the formula and get:

• \( y - 6 = -\frac{4}{3}(x + 3) \)

This equation directly shows how the line behaves around the point.Using the point-slope form can make it easier to derive or manipulate the equation further, especially if you need to convert it into a different form, such as the slope-intercept form.

Slope-Intercept Form

The slope-intercept form is perhaps the most recognized equation of a line. It’s typically used because it directly tells you the slope and the y-intercept of the line. This form is expressed as:

• \( y = mx + b \)

Here, \( m \) is the slope, and \( b \) is the y-intercept, where the line crosses the y-axis. This representation of a line makes it easy to quickly sketch the graph of the line or understand changes in the slope or intercept.
From our exercise, converting the point-slope form \( y - 6 = -\frac{4}{3}(x + 3) \) to slope-intercept involves some algebraic rearrangement.
Solving for \( y \), we end with:

• \( y = -\frac{4}{3}x + 2 \)

The y-intercept, \( b = 2 \), reveals exactly where the line crosses the y-axis, and the slope \( m = -\frac{4}{3} \) shows the line’s steepness and direction.

Calculating Slope

Finding the slope of a line connecting two points ensures you understand the line’s steepness and how it inclines or declines. The slope (often represented by \( m \)) is calculated by taking the difference in the y-values of two points and dividing by the difference in the x-values. This process is expressed by the formula:

• \( m = \frac{y_2-y_1}{x_2-x_1} \)

Using this formula makes finding the slope straightforward. So with the points \((-3, 6)\) and \((3, -2)\), the calculation becomes:

• \( m = \frac{-2 - 6}{3 - (-3)} = \frac{-8}{6} = -\frac{4}{3} \)

This slope \( -\frac{4}{3} \) informs us about the line’s direction: negative slopes indicate the line is decreasing, moving from left to right.
Understanding how to calculate the slope is essential for interpreting linear relationships in graphs, enhancing both your skills in graphing and analyzing data.