Y-Intercept: Solving Y - 6 = -3/2(x - 4) Simply

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Finding the Y-Intercept of the Line y - 6 = -3/2(x - 4)

Hey guys! Today, we're diving into a super common algebra problem: finding the y-intercept of a line. Specifically, we’re going to tackle the equation y - 6 = -3/2(x - 4). Don't worry; it's easier than it looks! We'll break it down step by step so you can master these types of problems. Understanding the y-intercept is crucial in various mathematical contexts, from graphing linear equations to solving real-world problems. The y-intercept is the point where the line crosses the y-axis, and it's represented by the coordinate (0, b), where b is the y-value. Let’s get started and make sure you understand exactly how to find it.

Understanding the Y-Intercept

Before we jump into solving the problem, let’s make sure we’re all on the same page about what the y-intercept actually is. The y-intercept is simply the point where a line crosses the y-axis on a graph. Think of it as the spot where the line 'intercepts' the y-axis. This happens when the x-value is zero. So, to find the y-intercept, we need to figure out what the y-value is when x is zero. This concept is fundamental in understanding linear equations and their graphical representations. Recognizing that the y-intercept occurs when x = 0 is the key to solving these problems. The y-intercept is often denoted as b in the slope-intercept form of a linear equation (y = mx + b), where m is the slope and b is the y-intercept. Grasping this concept will make the following steps much clearer and help you solve similar problems with confidence. Now that we have a solid understanding of the y-intercept, let's move on to solving our specific equation.

Step-by-Step Solution

Okay, let's get into the nitty-gritty of solving our equation: y - 6 = -3/2(x - 4). Remember, our goal is to find the value of y when x is 0. Here’s how we'll do it:

1. Substitute x = 0

First things first, we're going to replace x with 0 in our equation. This is because, as we discussed, the y-intercept is the point where the line crosses the y-axis, and that happens when x is 0. So, our equation becomes:

y - 6 = -3/2(0 - 4)

2. Simplify the Equation

Now, let's simplify the equation. We'll start by simplifying the expression inside the parentheses:

y - 6 = -3/2(-4)

Next, we'll multiply -3/2 by -4. Remember that multiplying two negative numbers gives us a positive number:

y - 6 = 6

3. Isolate y

Our final step is to isolate y. To do this, we'll add 6 to both sides of the equation:

y - 6 + 6 = 6 + 6

This simplifies to:

y = 12

So, we’ve found that when x is 0, y is 12. This means the y-intercept is 12. Great job! You've just navigated through the key steps to finding the y-intercept.

Identifying the Correct Option

Alright, now that we've done the math and found that the y-intercept (b) is 12, let's look at our answer choices and identify the correct one. We were given these options:

A. b = -3/2 B. b = -6 C. b = 4 D. b = 12

We can clearly see that option D, b = 12, matches our solution. So, the correct answer is D. Make sure you always double-check your answer against the options provided. It’s easy to make a small mistake in the calculations, so verifying your result is a crucial step in problem-solving. By correctly identifying the y-intercept as 12, we’ve successfully solved the problem. Now, let's reinforce this learning with a recap and some extra tips.

Recap and Tips for Success

Awesome work, guys! Let's quickly recap what we’ve learned and then share some tips to help you nail these problems every time. Finding the y-intercept involves a few simple steps:

  1. Substitute x = 0 into the equation.
  2. Simplify the equation by performing the necessary arithmetic operations.
  3. Isolate y to find the y-intercept.

In our example, we plugged in x = 0 into the equation y - 6 = -3/2(x - 4), simplified, and found that y = 12. This gave us the y-intercept of 12.

Tips for Success

  • Always remember: The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0.
  • Double-check your work: Math errors can happen, so take a moment to review your calculations.
  • Simplify carefully: Pay close attention to signs (positive and negative) when simplifying equations.
  • Practice makes perfect: The more you practice these types of problems, the easier they become.
  • Visualize the problem: If possible, try to visualize the line and where it might cross the y-axis. This can help you check if your answer makes sense.

With these tips in mind, you’ll be well-equipped to tackle any y-intercept problem that comes your way. Remember, understanding these fundamental concepts is key to mastering algebra and other math topics. So keep practicing, and you’ll become a pro in no time!

Why Understanding Y-Intercepts is Important

Okay, so we've figured out how to find the y-intercept, but let's chat a bit about why it's so important. Knowing the y-intercept isn't just a cool math trick; it actually helps us understand and interpret linear equations in a real-world context. The y-intercept represents the value of y when x is zero. Think about it this way: in many real-life scenarios, the y-axis represents a starting point or initial value. For example, if you're tracking the growth of a plant over time, the y-intercept might represent the plant's height when you first planted it (time = 0). Or, if you're looking at a graph of a car's depreciation, the y-intercept could be the initial value of the car when it was brand new. Understanding these real-world applications makes learning about y-intercepts way more engaging and helps you see the bigger picture. Plus, when you can connect math concepts to everyday situations, they stick with you better. So, next time you're solving for a y-intercept, remember that you're not just crunching numbers; you're uncovering a crucial piece of information that can tell a story.

Practice Problems

Now that you've got the hang of finding the y-intercept, let’s put your skills to the test with a few practice problems. Working through these will solidify your understanding and help you feel even more confident. Remember the steps we discussed: substitute x = 0, simplify the equation, and isolate y. Grab a piece of paper and a pencil, and let’s get started!

  1. Find the y-intercept of the line y = 2x + 5.
  2. What is the y-intercept of the equation y - 3 = -(x + 1)?
  3. Determine the y-intercept for the line y + 4 = 3/4(x - 8).

Take your time and work through each problem carefully. Don’t be afraid to go back and review the steps we covered earlier if you need a refresher. Once you’ve solved each problem, double-check your answers to make sure they make sense. If you get stuck, try breaking the problem down into smaller steps or visualizing the line on a graph. Remember, practice is key to mastering any math skill, so the more problems you solve, the better you’ll become. Keep challenging yourself, and you’ll be a y-intercept pro in no time! Good luck, and happy solving!