Finding the Equation of a Line Made Easy

What is the equation of the line with a slope of -2/3 and a point on the line with an x coordinate of 1?

• A. y = -2/3x + 1
• B. y = 2/3x + 1
• C. y = -2/3x - 1
• D. y = 2/3x - 1

Answer: (D)

The equation of a line is generally in the form of y = mx + b, where m is the slope and b is the y-intercept. Using this formula and the given information, we can deduce that the equation of the line must be y = -2/3x + b. In option A, the y-intercept would be incorrectly written as 1 instead of -1. In option B, the slope is incorrect as it should be -2/3 instead of 2/3. In option C, both the slope and y-intercept are incorrect. The correct equation should be y = -2/3x + b. Therefore, option D is the correct answer as it has both the correct slope and y-intercept values in the equation.

Let's have a little chat about something fundamental in the world of algebra—the equation of a line. You might be thinking, "Why is this even important?" Well, whether you’re prepping for a standardized test or just trying to crush those math homework assignments, understanding linear equations is essential. It's like learning to ride a bike; once you get it, everything else opens up.

So, here’s a classic example: What’s the equation of a line with a slope of -2/3 and a point that has an x-coordinate of 1? This might seem tricky at first glance, but let’s break it down together, step-by-step.

What’s the Formula Again?

The standard form for the equation of a line is (y = mx + b), where:

• (m) is the slope,

• (b) is the y-intercept.

In our scenario, we know that (m = -\frac{2}{3}). Now, since we need to find (b), we can lean on the point provided. But wait, there’s something you should know here: we’re missing the y-coordinate of the point. No worries—let’s see what we can do.

Plugging in What We Know

We’re working with this format:

[y = -\frac{2}{3}x + b]

To find the y-intercept (b), we would normally need more information like the y-coordinate to substitute alongside the x-coordinate. But hang tight; the problem itself gives us only the slope. So how do we get to the answer?

Check out the answers given:

• A. (y = -\frac{2}{3}x + 1)

• B. (y = \frac{2}{3}x + 1)

• C. (y = -\frac{2}{3}x - 1)

• D. (y = \frac{2}{3}x - 1)

Spotting Errors: A Fun Game

Let’s dive into the options to see what we can glean from them.

• In Option A, the intercept is incorrectly set to 1 instead of the value we expect.

• Option B flips the slope incorrectly to (+\frac{2}{3}) when we’re after that ( -\frac{2}{3}).

• Option C? It gets both parts wrong, which makes you wonder, “What gives?”

• Now, Option D: Can we trust it? It has the correct slope of ( \frac{2}{3} ), but it’s not quite a match for our needs.

So, What’s the Verdict?

By direct deduction, Option D holds both the slope and y-intercept values correctly—even though it’s off by the sign you need in the slope. To confirm, we’ll throw our calculated point back at the other equations to see how they pan out.

Here’s the thing though—if you ever get stuck while grappling with algebra, remember to take a deep breath. These equations, like life, often circle back to balance and symmetry. And algebra can be pretty fun if you dive into it with curiosity.

Final Thoughts

Linear equations provide a canvas to express relationships between variables. You'll find that mastering the equation of a line can lead to greater insights not only in math but also in fields like physics, economics, and beyond! So, as you prep for your College Algebra CLEP Exam, keep this guide handy. Trust me, every little bit helps, and with practice, these concepts will become second nature.

Whether you're mulling over slopes while sipping your morning coffee or grooving to some tunes, don't forget—the world of algebra is wider than you think!