Mastering Slope and Points: Your Path to College Math Success

Which point lies on the line passing through (4, 5) with a slope of -2?

• A. (4, 1)
• B. (6, 2)
• C. (5, 3)
• D. (5, 7)

Answer: (C)

To determine which point lies on the line passing through the point (4, 5) with a slope of -2, we can use the point-slope form of a linear equation. The equation can be written as follows: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. Substituting in our values: y - 5 = -2(x - 4). Now, we simplify this equation to find the line's equation: y - 5 = -2x + 8, y = -2x + 13. This is the equation of the line. Next, we can test each of the points given in the options to see if they satisfy this equation. For the point (5, 3): Substituting x = 5 into the equation: y = -2(5) + 13, y = -10 + 13, y = 3. Since the calculated y-value matches the y-coordinate of the point (5, 3), this point lies on the line. To verify the other points: - For (4, 1): Plugging in x

Are you gearing up for the College Math Placement Test and feeling a bit anxious about the equations? Don’t sweat it! Let’s break down one fundamental concept: determining which point lies on a line given a specific slope and a point on that line. By the end of this article, you’ll feel more confident dealing with slopes and lines—so grab a pencil and let’s jump in!

Imagine we’re given a line that passes through the point (4, 5) and has a slope of -2. Your task? Find out which point from a list lies on that line. Sounds tricky? It’s actually easier than it looks!

To start, we’ll use the point-slope form of a linear equation, which looks like this:

y - y₁ = m(x - x₁)

In this formula, (x₁,y₁) is a point on the line (ours is (4, 5)), and m is the slope (-2 in our case). Substituting in these values, we get:

y - 5 = -2(x - 4)

Now, let’s simplify that equation to write it in slope-intercept form, which is typically written as y = mx + b. Here’s how it unfolds:

1. Rearranging gives us:

[

y - 5 = -2x + 8

]

2. Adding 5 to both sides:

[

y = -2x + 13

]

And there you have it—the equation of the line! You might be thinking, “Great, but how do I find out which point lies on this line?” No problem! We’ll test each of the points given in your options to see which one fits.

Let’s Check Each Point!

• For (5, 3): Plug x = 5 into the equation we just found.

[

y = -2(5) + 13

]

[

y = -10 + 13

]

[

y = 3

]

Bingo! This matches the y-coordinate of (5, 3). So, it lies on the line.

• Now, let’s verify the others:

• For (4, 1):

[

y = -2(4) + 13 = -8 + 13 = 5 \quad \text{(not a match)}

]

• For (6, 2):

[

y = -2(6) + 13 = -12 + 13 = 1 \quad \text{(not a match)}

]

• For (5, 7):

[

y = -2(5) + 13 = -10 + 13 = 3 \quad \text{(not a match)}

]

So What’s the Takeaway?

Through just a bit of algebra, we see that (5, 3) is the point that lies on the line for the slope -2 through (4, 5). This skill is not only vital for acing your College Math Placement Test; it’s also super handy for real-world situations, such as when you're trying to find a trend line in data.

You know what? Understanding these principles can make you feel like a math wizard, and let's be honest, who wouldn’t want that? Keep practicing, and in no time, these concepts will become second nature. Math doesn't have to be daunting; it can be like pie—easy and delightful with the right slice! So, are you ready to tackle your test with newfound confidence?

Take a deep breath and let's keep pushing those boundaries in your math journey. You got this!