Problem 15 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((4,8)\) and \((0,4)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = x + 4 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

Step 2: Identify and Label Points

We have two points given: \((4, 8)\) and \((0, 4)\). Here, \((x_1, y_1) = (4, 8)\) and \((x_2, y_2) = (0, 4)\).

Step 3: Calculate the Slope

The formula for the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the values: \( m = \frac{4 - 8}{0 - 4} = \frac{-4}{-4} = 1 \). So, the slope \( m = 1 \).

Step 4: Determine the y-intercept

Since one point is \((0, 4)\), this directly gives the y-intercept, \( b = 4 \), because when \( x = 0 \), \( y = b \).

Step 5: Write the Equation in Slope-Intercept Form

Now substitute the slope \( m = 1 \) and y-intercept \( b = 4 \) into the equation \( y = mx + b \). Thus, the equation is \( y = 1x + 4 \) or simply \( y = x + 4 \).

Key Concepts

Equation of a Line

An equation of a line is a mathematical representation showing the relationship between the x and y coordinates on a two-dimensional graph. There are different forms of line equations, but the most widely used is the slope-intercept form. The slope-intercept form is expressed as \( y = mx + b \). This form is particularly handy because you can immediately identify the slope \( m \) and the y-intercept \( b \) from the equation. Being able to write the equation of a line allows you to understand how the line behaves and where it crosses the y-axis.

Calculation of Slope

The slope of a line, denoted by \( m \), is crucial for understanding how steep the line is. To calculate the slope when you have two points, use the formula:

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

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the given points.
This formula calculates the rate at which y increases or decreases as x increases. It essentially tells you how much y changes per unit increase in x. If the slope is positive, the line rises as you move to the right. Conversely, a negative slope means the line falls as you move to the right. In our exercise, utilizing the points \((4, 8)\) and \((0, 4)\), we calculated the slope as 1, indicating a consistent rise of 1 for every 1 unit increase in x.

y-intercept

The y-intercept of a line is the value of y where the line crosses the y-axis. In other words, it is the point on the graph where \( x = 0 \). In the slope-intercept form, \( y = mx + b \), the y-intercept is represented by \( b \).
This is an important characteristic because it provides a starting point from which the graph of the line can be easily drawn.
In the specific problem we solved, one of our points was \((0, 4)\), revealing that the y-intercept \( b \) is 4. This means that even before predicting the behavior of the line with slope, you know the line starts at 4 on the y-axis when x is zero.