Master the Equation of a Line: Unlocking the Mystery of Slope and Points

What is the equation of a line with a slope of 2 and passes through the point (5,21)?

• A. y = 5x + 21
• B. y = 2x + 21
• C. y = 5x - 21
• D. y = 2x - 21

Answer: (B)

A This equation does not have a slope of 2, as the coefficient of x is 5, not 2. C: This equation does not have a y-intercept of 21, as the constant term is -21, not 21. D: This equation does not have both a slope of 2 and a y-intercept of 21. The constant term is -21, not 21.

When you're studying for the College Math CLEP exam, things can sometimes feel like a whirlwind! One topic you’re probably facing is how to find the equation of a straight line given a slope and a point. It sounds intimidating, but once you get the hang of it, it’s actually a piece of cake! So let’s break it down together.

Riding the Slope

First things first. Let's talk about what it means to have a slope. Slope essentially tells us how steep a line is. A slope of 2 means that for every 1 unit you move horizontally, you rise 2 units vertically. Picture a staircase—each step up is two steps forward. Cool, right? Now, we’ve got a point at (5, 21), which means that when x is 5, y is 21. Can you picture that on a coordinate plane?

Here’s the general formula for the equation of a line:

y = mx + b

Where:

• m is the slope (in this case, 2)

• b is the y-intercept (the point where the line crosses the y-axis)

Let’s Plug In Those Numbers

From the slope, we already know (m = 2). Now we need to find that elusive y-intercept, (b). To find (b), we can plug our known point (5, 21) into the equation. So, let’s set things up:

[ 21 = 2(5) + b ]

Here’s the moment of truth—let’s calculate. What’s (2 \times 5)? That’s right, 10! So we have:

[ 21 = 10 + b ]

To isolate (b), we can subtract 10 from both sides. Now it’s becoming a little clearer:

[ b = 21 - 10 = 11 ]

And there we have it! Our y-intercept is 11. This means we cross the y-axis at 11. Now we can write the final equation of our line:

[ y = 2x + 11 ]

Let’s Check the Answers

Now, let’s take a look at the options given in the question:

A. y = 5x + 21

B. y = 2x + 21

C. y = 5x - 21

D. y = 2x - 21

The correct answer, as we just figured out, is B: y = 2x + 21! Wait—hold on, that was actually a little tricky. Upon closer inspection, it seems we got sidetracked; this equation should have a y-intercept of 11 and not 21! So here’s a reminder: check the details carefully!

The true equation is: y = 2x + 11. Somehow, it’s easy to let those little details slip by when you're juggling everything for the exam, isn’t it?

Wrap Up with These Reminders

Understanding the relationship between slope and points on a graph will make your math journey a whole lot easier and more enjoyable. Whether you visualize it with real-world objects like a growing ladder or use graph paper to sketch it out, engaging with the material is key!

So remember, when you encounter a problem involving the equation of a line, pause for a second, identify your pieces (slope and points), and plug them into that handy formula. You got this! Stay strong with your CLEP exam prep—every equation conquered is a step closer to your goals!