Problem 38 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. \(x\) -intercept \(=4\) and \(y\) -intercept \(=-2\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = 0.5(x-4)\) and in slope-intercept form is \(y = 0.5x - 2\)

Step 1: Calculate the Slope

The formula to find the slope (m) given two points (x1,y1) and (x2,y2) is: \(m = \frac{y2 - y1}{x2 - x1}\). In this case, the two points are the x and y intercepts: (4,0) and (0,-2). When these points are inserted into the slope equation, the slope (m) equals: \[m = \frac{-2 - 0}{0 - 4} = 0.5\]

Step 2: Write the equation in point-slope form

The point-slope form of a line is \(y - y1 = m(x - x1)\). Replacing \(m = 0.5\) and choosing any of the two points, preferentially the one with the least complex numbers, let's choose (4,0), the point-slope form of the line will be: \(y - 0 = 0.5(x - 4)\)

Step 3: Write the equation in slope-intercept form

The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The line can be rewritten from point-slope form to get it into slope-intercept form: \[y = 0.5x -2\].

Key Concepts

Understanding Point-Slope Form

The point-slope form is a way to represent the equation of a line using the slope and a single point on the line. It's written as:

• \( y - y_1 = m(x - x_1) \)

Where:

• \( m \) is the slope.
• \( (x_1, y_1) \) is a point on the line.

To use this form, you need the slope of the line, which tells you how steep the line is. You also need a point through which the line passes. If given two different points, you can find the slope first, then pick one of the points to use this form.
For example, if we have a line that passes through the point (4,0) with a slope of 0.5, the point-slope form becomes:
\( y - 0 = 0.5(x - 4) \).
This form is particularly useful for quickly forming an equation when both the slope and a point are known.

Exploring Slope-Intercept Form

The slope-intercept form of a line is a straightforward and popular way to write a line's equation. It is represented as:

• \( y = mx + b \)

Where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

This form is widely used because it immediately reveals both the slope and the y-intercept of the line. This makes it easier to graph or understand the line's general behavior. Converting from point-slope form to slope-intercept form involves solving the equation for \( y \).
In our example, transforming the equation \( y - 0 = 0.5(x - 4) \) to the slope-intercept form gives us \( y = 0.5x - 2 \), highlighting both the slope and where it crosses the y-axis.

Understanding Intercepts

Intercepts are the points where a line crosses the axes. They are helpful in understanding and graphing the behavior of lines.**X-Intercept**The x-intercept is where the line crosses the x-axis. At this point:

• The value of \( y \) is zero.

You can find it by setting \( y = 0 \) in the line equation and solving for \( x \). In this exercise, the x-intercept is 4.
**Y-Intercept**Similarly, the y-intercept is where the line crosses the y-axis. Here:

• The value of \( x \) is zero.

This is found by setting \( x = 0 \) in the equation and solving for \( y \). The given y-intercept is -2 in this case.
Knowing the intercepts gives you solid foundation points while graphing a line, making it easier to see the line's position and slope.