Understanding the Slope: A Key Concept in College Algebra

What is the slope of the line passing through the points (2, 3) and (5, 8)?

• A. -3
• B. 3
• C. 1/3
• D. -1/3

Answer: (B)

The slope, also known as the gradient, is a measure of how steep a line is. It is calculated by dividing the change in the y-coordinates by the change in the x-coordinates between two points on a line. In this case, we can see that for the given points, there is an increase of 5 in the y-coordinate and an increase of 3 in the x-coordinate. This means that the slope of the line is 5/3. Since the given options for the slope are all whole numbers or fractions, we can eliminate options A and D which are negative. This is because the given points have a positive slope. Additionally, option C of 1/3 is incorrect because as we can see, the slope is larger than 1. Therefore, the only correct option is B which has a slope of 5/3.

When you think about lines on a graph, do you ever get curious about their steepness? The slope of a line is more than just a number; it’s a fundamental concept in College Algebra that tells us how steep a line is. Imagine you're climbing a hill; some slopes make you feel like a mountain goat, while others are a gentle stroll in the park.

Let’s take a look at how we can calculate the slope of a line between two points. Picture two points on a graph—let’s say (2, 3) and (5, 8). How do we find the slope? Here’s the scoop: the slope (often represented by 'm') is essentially the change in y divided by the change in x. So, we’re looking at how much ‘y’ changes as we move along ‘x.’

So, what’s the math here? First, we need to identify our coordinates:

• Point A (2, 3): Here, 2 is the x-coordinate and 3 is the y-coordinate.

• Point B (5, 8): In this case, 5 is again our x-value, and 8 our y-value.

Now, let’s compute the changes:

1. Calculate the change in y (Δy): 8 (from Point B) - 3 (from Point A) = 5

2. Calculate the change in x (Δx): 5 (from Point B) - 2 (from Point A) = 3

Now, we can plug these values into our slope formula:

[

m = \frac{Δy}{Δx} = \frac{5}{3}

]

Okay, let’s backtrack for a moment—how do we know this slope is positive? Great question! Look at our points again. Because both changes in ‘y’ and ‘x’ are positive, we end up with a positive slope. Can you see how it’s greater than 1? That helps us eliminate options A (-3) and D (-1/3) right off the bat since they’re negative. Also, can we rule out option C (1/3)? Absolutely! The slope of 5/3 indicates a much steeper incline than option C suggests.

So, we're left with option B. Voilà! The slope of the line that passes through the points (2, 3) and (5, 8) is indeed 5/3.

Now, isn’t that fascinating? When you break it down, slope isn’t just a boring number; it reflects relationships in data, helping us understand trends or changes over time.

As you prep for your College Algebra CLEP test, remember to take the time to grasp these concepts rather than just memorize them. After all, math isn’t just about the answer; it’s about understanding the journey to get there. More importantly, the next time someone asks about slopes, you can confidently throw down your knowledge like a pro! So, ready to tackle those algebra fears? Let’s do this!