Problem 24 - College Algebra

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. $$ x \text { -intercept is } 1, \text { and }(-2,6) $$

Step-by-Step Solution

Answer

The slope-intercept form of the line is \( y = -2x + 2 \).

Step 1: Determine the x-intercept

The x-intercept of the line gives us a point on the line, specifically where the line crosses the x-axis. This point has coordinates (1, 0).

Step 2: Determine the slope (m)

The slope of the line, \(m\), can be calculated using the two points given: the x-intercept (1, 0) and the point (-2, 6). The formula for slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in the coordinates of the points, we have: \[ m = \frac{6 - 0}{-2 - 1} = \frac{6}{-3} = -2 \] So, the slope \(m = -2\).

Step 3: Use Point-Slope Form to Write the Equation

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \). We can use the point (-2, 6) and the slope \(m = -2\) calculated in the previous step: \[ y - 6 = -2(x + 2) \] Simplifying this gives \( y - 6 = -2x - 4 \).

Step 4: Convert to Slope-Intercept Form

The slope-intercept form of a line's equation is \(y = mx + b\). From the equation \( y - 6 = -2x - 4 \), adding 6 to both sides gives: \[ y = -2x + 2 \] So, the slope-intercept form of the line is \( y = -2x + 2 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most friendly ways to express a line. It is written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept, which tells us where the line crosses the y-axis. This form is handy because it gives a clear picture of how the line behaves:

• Slope \( (m) \): Indicates the steepness and direction of the line.
• Y-intercept \( (b) \): The point where the line touches the y-axis, making the connection to the graph more intuitive.

For example, for the equation \( y = -2x + 2 \), the slope \( m \) is -2, showing a downward slope, and the y-intercept \( b \) is 2, crossing the y-axis at the point (0, 2). By using slope-intercept form, you can quickly sketch the graph of a line starting at \( b \) and following the direction indicated by \( m \).
Overall, converting equations to this form can simplify the process of graphing and understanding linear characteristics.

X-intercept

The x-intercept is where the line crosses the x-axis, and it can be very useful for understanding how the line interacts with the coordinate plane. To identify the x-intercept, set \( y = 0 \) in the equation, because at this point the line has not moved up or down from its initial horizontal position.

• Coordinates: The x-intercept has coordinates \((x, 0)\).
• Context: In this exercise, the x-intercept is given as \( x = 1 \), or the point (1, 0), helping to anchor our line graphically.

If you have the slope-intercept form of a line \( y = -2x + 2 \), you can find the x-intercept by solving \( 0 = -2x + 2 \). Simply solve the equation for \( x \):
\[ -2x + 2 = 0 \]
\[ 2 = 2x \]
\[ x = 1 \]
Here, the calculated x-intercept is (1, 0), which corresponds to the given in the exercise and precisely represents where the line touches the x-axis.

Slope Calculation

The slope of a line is a measure of its steepness. It describes how quickly it moves up or down as it travels from left to right. Calculating the slope involves understanding the change in the y-coordinate (rise) over the change in the x-coordinate (run). The formula to determine the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Using the given points from the exercise, (1, 0) and (-2, 6), the slope is calculated as follows:

• First, identify the differences:
  \( y_2 - y_1 = 6 - 0 = 6 \)
• Then, \( x_2 - x_1 = -2 - 1 = -3 \)
• Now, substitute into the formula:
  \( m = \frac{6}{-3} = -2 \)

This slope of \( -2 \) indicates a line that falls as it moves from left to right. Each step horizontally to the right, it drops two units vertically. Understanding how to calculate and interpret slope is essential for assessing the nature and direction of any linear relationship.