Finding X And Y Intercepts: A Step-by-Step Guide

Hey everyone! Today, we're diving into a super important concept in algebra: finding the x- and y-intercepts of a line. This is a fundamental skill, guys, and it's used all over the place in math and real-world applications. Trust me, understanding intercepts is going to make your life a whole lot easier when dealing with linear equations and graphing. So, let's get started, and I'll break it down in a way that's easy to understand. We'll find out the correct answer of the given question. Let's go!

What are x- and y-intercepts?

Before we jump into the problem, let's make sure we're all on the same page about what x- and y-intercepts actually are. The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. Think of it this way: if you're walking along the x-axis, you're not going up or down (no y movement). The y-intercept, on the other hand, is the point where the line crosses the y-axis. Here, the x-coordinate is always zero. This is where the line hits the vertical axis. Understanding these definitions is super crucial for solving the problem effectively. The x-intercept is the point (x, 0), and the y-intercept is the point (0, y). Got it? Awesome! Now, let's get our hands dirty with an example.

Now, let's explore how to find these intercepts. The process is pretty straightforward, but it's important to follow the steps correctly to avoid any confusion. We'll go through the given linear equation and find the x and y intercepts. In simple words, the x-intercept is where the line cuts the x-axis (where y = 0), and the y-intercept is where the line cuts the y-axis (where x = 0). It’s all about finding those specific points where the line meets the axes. Keep in mind that these intercepts are super helpful when you're graphing linear equations. They give you two points that you can use to draw the line accurately.

Understanding the Concepts

Let’s make sure we've got the basics down before diving into calculations. Understanding the x- and y-axes is like understanding the foundation of a house. The x-axis is the horizontal line, and the y-axis is the vertical line. The point where they cross each other is called the origin (0, 0). The x-intercept is the point where a line intersects the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept, you set y = 0 in your equation and solve for x. The y-intercept is the point where a line intersects the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept, you set x = 0 in your equation and solve for y. This is the core of finding intercepts and making sense of linear equations. It's like having a key to unlock the secrets of a graph. Once you get the hang of it, you'll be finding intercepts like a pro, no sweat!

Solving the Problem: Find the x- and y-intercept of the line: -10x - 6y = 120

Alright, let's roll up our sleeves and solve the equation: -10x - 6y = 120. This is the fun part! We will find the x- and y-intercepts of the given line. Remember, the x-intercept is where the line crosses the x-axis (where y = 0), and the y-intercept is where the line crosses the y-axis (where x = 0). So, we'll use these facts to find our intercepts.

Finding the x-intercept

To find the x-intercept, we need to set y = 0 in the equation. So, let's substitute y with 0: -10x - 6(0) = 120. This simplifies to -10x = 120. Now, to solve for x, we divide both sides by -10: x = 120 / -10. This gives us x = -12. So, the x-intercept is -12. This means the line crosses the x-axis at the point (-12, 0). Easy peasy, right?

So, to get the x-intercept, we set y=0 in the equation -10x - 6y = 120. This becomes -10x - 6(0) = 120, which simplifies to -10x = 120. Dividing both sides by -10, we get x = -12. Therefore, the x-intercept is -12.

Finding the y-intercept

Now, let's find the y-intercept. To do this, we set x = 0 in the equation. So, we substitute x with 0: -10(0) - 6y = 120. This simplifies to -6y = 120. To solve for y, we divide both sides by -6: y = 120 / -6. This gives us y = -20. Therefore, the y-intercept is -20. This means the line crosses the y-axis at the point (0, -20). Fantastic, we did it!

To find the y-intercept, we set x=0 in the equation -10x - 6y = 120. This becomes -10(0) - 6y = 120, which simplifies to -6y = 120. Dividing both sides by -6, we get y = -20. Hence, the y-intercept is -20.

Conclusion

So, guys, we have successfully found the x- and y-intercepts of the line -10x - 6y = 120. The x-intercept is -12, and the y-intercept is -20. This corresponds to option D in your question. Keep practicing, and you'll become a pro at finding intercepts. It's an essential skill in algebra that will help you visualize and understand linear equations more effectively. Remember, the key is to set the opposite variable to zero and solve for the remaining variable. That's all there is to it!

In summary, the x-intercept is -12 and the y-intercept is -20. Option D, which states “x-intercept is -12; y-intercept is -20,” is the correct answer. Congratulations, you've mastered another algebra concept! Keep practicing and you will do great.

Answer

Based on our calculations:

A. x-intercept is -20; y-intercept is -12.

B. x-intercept is -10; y-intercept is -6.

C. x-intercept is -6; y-intercept is -10.

D. x-intercept is -12; y-intercept is -20.

The correct answer is D.