Problem 8 - College Algebra with Modeling and Visualization

Question

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Passing through \((-6,10)\) and \((5,-12)\)

Step-by-Step Solution

Answer

The point-slope form is \( y - 10 = -2(x + 6) \) and the slope-intercept form is \( y = -2x - 2 \).

Step 1: Calculate the Slope

To find the slope \( m \) of the line passing through the points \((-6, 10)\) and \((5, -12)\), we'll use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plug in the values \( y_2 = -12 \), \( y_1 = 10 \), \( x_2 = 5 \), and \( x_1 = -6 \).\[ m = \frac{-12 - 10}{5 - (-6)} = \frac{-22}{11} = -2 \] So, the slope \( m \) is \(-2\).

Step 2: Write the Point-Slope Form

Using the point-slope form formula \( y - y_1 = m(x - x_1) \), we choose the first point \((-6, 10)\) and the slope \( m = -2 \): \[ y - 10 = -2(x + 6) \] This is the equation in point-slope form.

Step 3: Distribute and Simplify

Expand the equation from Step 2: \[ y - 10 = -2(x + 6) \] Distribute the \(-2\) to get: \[ y - 10 = -2x - 12 \]

Step 4: Convert to Slope-Intercept Form

Add \(10\) to both sides of the equation from Step 3 to solve for \(y\): \[ y = -2x - 12 + 10 \] Simplify the right side: \[ y = -2x - 2 \] This is the equation in slope-intercept form, \( y = mx + b \), where \( m = -2 \) and \( b = -2 \).

Key Concepts

Slope Calculation

The slope of a line, often represented by the letter \( m \), is a measure of its steepness and direction. When given two points on a plane such as \((-6, 10)\) and \((5, -12)\), you can calculate the slope using the slope formula:

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

For our points, substitute \( y_2 = -12 \), \( y_1 = 10 \), \( x_2 = 5 \), and \( x_1 = -6 \), resulting in:

• \( m = \frac{-12 - 10}{5 - (-6)} \)
• Simplifying gives \( m = \frac{-22}{11} \)
• The slope \( m = -2 \)

The negative sign indicates that the line falls as it moves from left to right, confirming the direction and nature of the line. The slope is crucial in defining the line's equation and understanding its characteristics. Remember, a positive slope means the line rises, while a negative slope means it falls. A slope of zero indicates a horizontal line, and an undefined slope suggests a vertical line.

Line Equations

Line equations are mathematical representations that describe straight lines on a coordinate plane. There are several forms, but point-slope form is particularly useful when you have a point on the line and the slope. The point-slope form equation is given by:

• \( y - y_1 = m(x - x_1) \)

In our exercise, substituting the slope \( m = -2 \) and the point \((-6, 10)\), we have:

• \( y - 10 = -2(x + 6) \)

This form allows you to see the starting point of the line and how the line moves with the slope. This representation is powerful because you can quickly write the equation of a line if you know any point on the line and its slope. It also serves as a stepping stone to convert into other forms of line equations, which may be simpler for different purposes.

Slope-Intercept Form

The slope-intercept form of a line is one of the most common and useful representations for quickly understanding a line's characteristics. It is structured as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

To convert from point-slope to slope-intercept form, you start by distributing and simplifying the point-slope equation:

• Begin with the point-slope form: \( y - 10 = -2(x + 6) \)
• Distribute the \(-2\): \( y - 10 = -2x - 12 \)
• Add 10 to both sides to isolate \( y \): \( y = -2x - 12 + 10 \)
• Simplifying gives the slope-intercept form: \( y = -2x - 2 \)

The coefficient of \( x \), which is \(-2\), confirms the consistent slope we calculated. The constant \( -2 \) represents the y-intercept, which is where the line crosses the y-axis. This form allows you to quickly graph the line or understand its position and movement on the plane at a glance.