X And Y Intercepts: Solving -6x + 9y = -18

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X and Y Intercepts: Solving -6x + 9y = -18

Hey guys! Today, we're going to tackle a common algebra problem: finding the x and y-intercepts of a linear equation. Specifically, we'll be working with the equation -6x + 9y = -18. Don't worry, it's not as scary as it looks! We'll break it down step by step, so you can master this skill and impress your friends (or at least ace your next math test!). Understanding intercepts is super important for graphing linear equations and understanding their behavior. So, let’s dive in and make math a little less mysterious, shall we?

Understanding Intercepts

Before we jump into solving the equation, let's make sure we're all on the same page about what intercepts actually are. Think of them as the points where a line crosses the x and y-axes on a graph.

  • The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. So, the x-intercept is written as (x, 0).
  • The y-intercept is the point where the line crosses the y-axis. Similarly, at this point, the x-coordinate is always zero. The y-intercept is written as (0, y).

Why are intercepts important? Well, they give us two key points that we can use to easily graph a line. Remember, you only need two points to define a line! Plus, intercepts often have real-world meaning in applied problems, representing starting values or break-even points.

Step-by-Step Solution

Okay, now that we know what intercepts are, let's get down to business and find them for the equation -6x + 9y = -18. We'll find the x-intercept first, then the y-intercept. It's a pretty straightforward process, so stick with me!

Finding the x-intercept

To find the x-intercept, we need to remember that the y-coordinate is zero at this point. So, we're going to substitute y = 0 into our equation and solve for x. Here's how it looks:

  1. Substitute y = 0: -6x + 9(0) = -18
  2. Simplify: -6x + 0 = -18 -6x = -18
  3. Divide both sides by -6: x = -18 / -6 x = 3

So, the x-intercept is (3, 0). That means the line crosses the x-axis at the point where x is 3.

Finding the y-intercept

Now, let's find the y-intercept. This time, we'll substitute x = 0 into our equation and solve for y, since the x-coordinate is zero at the y-intercept. Let's do it:

  1. Substitute x = 0: -6(0) + 9y = -18
  2. Simplify: 0 + 9y = -18 9y = -18
  3. Divide both sides by 9: y = -18 / 9 y = -2

Therefore, the y-intercept is (0, -2). This means the line crosses the y-axis at the point where y is -2.

Putting It All Together

Great job, guys! We've successfully found both the x and y-intercepts of the equation -6x + 9y = -18. To recap:

  • The x-intercept is (3, 0).
  • The y-intercept is (0, -2).

Now, if you wanted to graph this line, you could simply plot these two points on a coordinate plane and draw a straight line through them. Easy peasy!

Graphing the Line Using Intercepts

Okay, so we've found the intercepts, but how does that actually help us visualize the line? Well, as we mentioned earlier, two points are all you need to draw a straight line. The intercepts give us those two points directly! Let's walk through how to graph the line using the intercepts we just found.

  1. Plot the intercepts: On a coordinate plane, find the point (3, 0) and mark it. This is your x-intercept. Then, find the point (0, -2) and mark it. This is your y-intercept.

  2. Draw a line: Take a ruler or straightedge and carefully draw a straight line that passes through both of the points you just plotted. Make sure the line extends beyond the points in both directions.

  3. Check your work: A good way to double-check your graph is to pick another point on the line and see if it satisfies the original equation. For example, you could look at the point where x = 1 on your graph. It looks like the y-value is approximately -4/9. Let's plug x=1 into the equation and solve for y.

    -6(1) + 9y = -18 -6 + 9y = -18 9y = -12 y = -12/9 = -4/3

Whoops! Our estimation from the graph was a little off. It's important to remember that visual estimations can have inaccuracies, and the true value is determined by the equation. So, while graphing using intercepts is convenient, always rely on the equation for precise calculations.

Alternative Methods for Finding Intercepts

While substituting 0 for x and y is the most common way to find intercepts, it's worth noting that there are other approaches, especially if you're familiar with slope-intercept form (y = mx + b). Let's briefly touch on an alternative method.

Using Slope-Intercept Form

  1. Convert to slope-intercept form: Rearrange the equation -6x + 9y = -18 to solve for y. This will put it in the form y = mx + b, where m is the slope and b is the y-intercept.

    9y = 6x - 18 y = (6/9)x - 2 y = (2/3)x - 2

  2. Identify the y-intercept: In the slope-intercept form, the y-intercept is simply the constant term, 'b'. In our case, b = -2, so the y-intercept is (0, -2). We found this earlier using the substitution method, so it's a good confirmation!

  3. Find the x-intercept (optional): If you've already converted to slope-intercept form, you can still find the x-intercept by setting y = 0 and solving for x. However, it's often just as easy to go back to the original equation and use the substitution method we discussed earlier.

While converting to slope-intercept form is super useful for understanding the slope and y-intercept, it’s not always the fastest method for solely finding intercepts. It’s more of a holistic approach to understanding the linear equation. For quickly pinpointing intercepts, sticking with the substitution method (setting x or y to zero) is often the most efficient way to go.

Common Mistakes to Avoid

Finding intercepts is a fundamental skill, but it's easy to make a few common mistakes along the way. Let's highlight some pitfalls to watch out for so you can avoid them.

  1. Forgetting to substitute zero: The biggest mistake is forgetting the core principle of finding intercepts: you're setting either x or y to zero. Make sure you're plugging in '0' for the correct variable when finding each intercept.
  2. Incorrectly solving the equation: Once you've substituted zero, you need to solve the resulting equation for the other variable. Double-check your algebra steps, especially when dealing with negative signs or fractions.
  3. Mixing up x and y: It's crucial to remember which coordinate is zero for each intercept. The x-intercept has a y-coordinate of zero (x, 0), and the y-intercept has an x-coordinate of zero (0, y). Mixing these up will lead to incorrect points.
  4. Not writing intercepts as ordered pairs: Intercepts are points on a graph, so they should always be expressed as ordered pairs (x, y). Just giving the x or y value isn't enough; you need to specify the point.
  5. Estimating inaccurately from a graph: We touched on this earlier, but it's worth reiterating. If you're given a graph and asked to find the intercepts, try to read the values as precisely as possible. However, remember that graphical estimations can be slightly off. Always double-check your answers by plugging them back into the equation if you can.

By being mindful of these common errors, you'll be well on your way to mastering the art of finding x and y-intercepts!

Why are Intercepts Important?

We've learned how to find intercepts, but let's take a step back and think about why they're actually important. Understanding the significance of intercepts can make the whole process feel less like a mathematical exercise and more like a tool for understanding the world around you.

  1. Graphing Linear Equations: As we've discussed, intercepts provide two key points that make graphing a line super easy. This is their most direct application. Instead of making a table of values and plotting several points, you can simply find the intercepts and draw a line through them.
  2. Real-World Applications: Intercepts often have meaningful interpretations in real-world scenarios modeled by linear equations. For example:
    • If you're modeling the cost of a service with a linear equation, the y-intercept might represent the fixed cost (the cost you pay even if you don't use the service), and the x-intercept might represent the break-even point (the number of units you need to sell to cover your costs).
    • In a distance-time graph, the y-intercept could represent your starting position, and the x-intercept could represent the time when you reach a certain destination.
  3. Understanding the Behavior of a Line: Intercepts, along with the slope, give you a good sense of how a line behaves. They tell you where the line crosses the axes and whether it's increasing or decreasing.
  4. Solving Systems of Equations: Intercepts can sometimes be helpful in visualizing and solving systems of linear equations. The point where two lines intersect (if they do) is the solution to the system, and understanding the intercepts can help you sketch the lines and estimate the solution.

In short, intercepts are more than just points on a graph; they're valuable pieces of information that can help you understand and interpret linear relationships in a variety of contexts.

Practice Problems

Alright, guys, now that we've covered the theory and the step-by-step process, it's time to put your knowledge to the test! Here are a few practice problems for you to try. Grab a pencil and paper, and let's see if you can find those intercepts!

  1. Find the x and y-intercepts of the equation: 2x - 5y = 10
  2. Determine the intercepts for the linear equation: y = 3x + 6
  3. Calculate the x and y-intercepts of: -4x + 8y = -24

(Answers will be provided at the end of this section)

Remember to follow the steps we discussed: substitute y = 0 to find the x-intercept, and substitute x = 0 to find the y-intercept. Be careful with your algebra, and don't forget to write your answers as ordered pairs!

If you get stuck, don't worry! Go back and review the steps, or check out the solutions below. The key is to practice, practice, practice. The more you work with these concepts, the more comfortable you'll become.

Solutions to Practice Problems

Okay, ready to check your answers? Here are the solutions to the practice problems:

  1. 2x - 5y = 10
    • x-intercept: (5, 0)
    • y-intercept: (0, -2)
  2. y = 3x + 6
    • x-intercept: (-2, 0)
    • y-intercept: (0, 6)
  3. -4x + 8y = -24
    • x-intercept: (6, 0)
    • y-intercept: (0, -3)

How did you do? If you got them all correct, awesome! You're well on your way to mastering intercepts. If you missed a few, that's totally okay too. Just take a look at your work, identify where you went wrong, and try again. Learning math is a process, and mistakes are a valuable part of that process.

Conclusion

So, there you have it, guys! We've journeyed through the world of x and y-intercepts, learning what they are, how to find them, and why they matter. We tackled the equation -6x + 9y = -18, worked through several examples, and even explored some common pitfalls to avoid. I hope this has been a helpful and informative experience for you all!

The key takeaway is that finding intercepts is a fundamental skill in algebra, and it opens the door to understanding and graphing linear equations. By mastering this concept, you'll not only improve your math skills but also gain a valuable tool for analyzing real-world situations.

Keep practicing, keep exploring, and most importantly, keep asking questions! Math is a fascinating subject, and there's always something new to learn. Until next time, happy solving!