Problem 1 - College Algebra

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. $$(4,7)\( and \)(8,10)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (4,7) and (8,10) is 0.75. The line rises.

Step 1: Use the slope formula

The slope between two points \( (x1, y1) \) and \( (x2, y2) \) is given by the equation \( m = (y2 - y1) / (x2 - x1) \). For the given points (4,7) and (8,10), plug these values into the equation to calculate the slope. It would therefore be \( m = (10 - 7) / (8 - 4) \)

Step 2: Simplify the expression

After plugging in the values, simplify the equation to find the slope. Therefore, \( m = 3 / 4 = 0.75 \).

Step 3: Determine the direction of the line

The direction of the line is determined by the sign of the slope. If the slope is positive, the line rises. If the slope is negative, the line falls. If the slope is undefined (i.e., the denominator in the slope calculation is zero), the line is vertical. If the slope is zero, the line is horizontal. Given our computed slope of 0.75, which is positive, the line rises.

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction when graphed on a coordinate plane. To find the slope between two points, we use the slope formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]
Where \( m \) is the slope, \( (x1, y1) \) and \( (x2, y2) \) are pairs of coordinates for two points on the line. Imagine you are hiking between two points on a hill; the slope tells you how steep the path is. Let's apply this to an example. For the points (4,7) and (8,10), we calculate the slope by subtracting the y-values and dividing by the difference in x-values:
\[ m = \frac{10 - 7}{8 - 4} = \frac{3}{4} \]
This gives us a slope of \( 0.75 \), meaning for every four steps to the right, you move up three steps, creating a gentle incline.

Horizontal and Vertical Lines

Understanding horizontal and vertical lines is crucial when studying the geometry of lines. Horizontal lines are flat and have a slope of zero because they do not rise or fall regardless of how far along the line you move. In the slope formula, this happens when the change in y is zero.
Conversely, vertical lines run straight up and down and have an undefined slope because the change in x is zero, so you're dividing by zero, which is mathematically undefined. This concept is just like trying to climb a perfectly vertical wall; it's so steep that its steepness cannot be measured with a standard slope.
In practical terms, if you have two points with the same x-value, such as (3,2) and (3,8), you're looking at a vertical line. If the y-values match, say (5,7) and (2,7), that's a horizontal line.

Direction of a Line

The slope of a line can also tell us about the direction of the line. If the slope is positive, as in our example with a slope of \( 0.75 \), the line rises from left to right. It's like going up a hill.
If the slope is negative, the line falls, which means it goes down as we move from left to right, like descending a slope. When we have a slope of zero, the line is horizontal, which is flat and doesn't rise or fall, similar to walking on a level path.
Finally, an undefined slope indicates a vertical line, which doesn't move left or right as it goes up or down. It's like looking at the edge of a building from the ground up. By looking at the slope, you can predict whether you're graphically going 'uphill', 'downhill', or on a 'flat road', or hitting an 'impassable wall'.