Equation Of Line Through (3,3) & (3,-3): Explained!

by Sebastian Müller 52 views

Hey guys! Let's dive into a fascinating problem in mathematics: finding the equation of a line that passes through two specific points, (3, 3) and (3, -3). This might seem a bit tricky at first, but don't worry, we'll break it down step by step. Understanding the equation of a line is super important in various fields, from engineering to computer graphics, so let's get started!

The Fundamentals of Linear Equations

Before we tackle the specific problem, let's quickly recap the basics of linear equations. A linear equation essentially represents a straight line on a graph. The most common form we use to express a linear equation is the slope-intercept form: y = mx + b. In this equation:

  • y represents the vertical coordinate.
  • x represents the horizontal coordinate.
  • m represents the slope of the line, which tells us how steep the line is.
  • b represents the y-intercept, which is the point where the line crosses the y-axis.

Another useful form is the point-slope form, which is y - y₁ = m(x - x₁). This form is particularly handy when we know the slope of the line and a point it passes through. Here, (x₁, y₁) are the coordinates of the known point.

Understanding these fundamental forms is crucial because they give us the tools to describe any straight line mathematically. The slope (m) is a key concept because it quantifies the line's inclination—how much y changes for every unit change in x. The y-intercept, on the other hand, anchors the line on the coordinate plane by telling us where it intersects the vertical axis. With these basics in mind, we're well-equipped to find the equation of our line!

Calculating the Slope

To find the equation of the line passing through (3, 3) and (3, -3), our first step is to determine the slope. Remember, the slope (m) is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁). This formula essentially tells us the change in y divided by the change in x between two points on the line.

Let's plug in our points (3, 3) and (3, -3) into the formula. We'll call (3, 3) our point 1, so x₁ = 3 and y₁ = 3. Similarly, (3, -3) will be our point 2, making x₂ = 3 and y₂ = -3. Now, let's substitute these values into the slope formula:

m = (-3 - 3) / (3 - 3)

This simplifies to:

m = -6 / 0

Uh oh! We've encountered something interesting here. Division by zero is undefined in mathematics. This tells us that the slope of our line is undefined. What does this mean for the line itself? Well, a line with an undefined slope is a vertical line. Vertical lines are special cases because they don't fit the typical y = mx + b mold. They have a different kind of equation, which we'll explore next.

Identifying the Equation

Since our slope is undefined, we know we're dealing with a vertical line. Vertical lines have a unique characteristic: their x-coordinate is the same for every point on the line. Think about it – if a line goes straight up and down, the horizontal position (x-coordinate) never changes. So, how does this help us find the equation?

Let's look at the points we're given: (3, 3) and (3, -3). Notice anything similar? That's right, both points have an x-coordinate of 3. This is the key! Since every point on the line has an x-coordinate of 3, the equation of the line is simply:

x = 3

This equation tells us that no matter what the y-coordinate is, the x-coordinate will always be 3. This perfectly describes a vertical line passing through the points (3, 3) and (3, -3). So, we've found our equation! It's not in the usual y = mx + b form, but it's a perfectly valid equation for a vertical line. This highlights an important point: not all lines can be expressed in the slope-intercept form, and vertical lines are a prime example.

Visualizing the Line

To really solidify our understanding, let's visualize the line. Imagine a coordinate plane with the x-axis and y-axis. Now, plot the points (3, 3) and (3, -3). You'll see that they are directly above and below each other. If you draw a line connecting these points, you'll get a vertical line that cuts straight through the x-axis at the point where x = 3.

This visual representation helps reinforce why the equation is x = 3. Every point on this line has an x-coordinate of 3, and the line extends infinitely upwards and downwards. This mental picture is a great way to check your work and ensure that the equation you've found makes sense in the context of the problem. Visualizing mathematical concepts can often make them much clearer and easier to remember, so it's a valuable tool to have in your arsenal.

Alternative Methods

While we've found the equation using the slope formula and recognizing the vertical line characteristic, let's briefly consider alternative methods. One approach might be to try using the point-slope form y - y₁ = m(x - x₁). However, since the slope is undefined, this form doesn't directly apply. We would quickly realize that any attempt to substitute the undefined slope would lead to an impasse.

Another method might involve trying to force the equation into the slope-intercept form y = mx + b. If we tried this, we'd again run into the problem of the undefined slope. No matter what value we try to substitute for m, we wouldn't be able to make the equation work for both points (3, 3) and (3, -3). These attempts, while ultimately unsuccessful, are valuable in demonstrating why the x = 3 equation is the only correct solution. They highlight the limitations of certain forms and the importance of recognizing special cases like vertical lines.

Conclusion

So, to wrap things up, the equation of the line that passes through the points (3, 3) and (3, -3) is x = 3. We arrived at this answer by calculating the slope, recognizing that it was undefined, and understanding that this indicates a vertical line. We then used the common x-coordinate of the points to determine the equation. Guys, understanding these concepts not only helps you solve this specific problem but also builds a solid foundation for more advanced topics in mathematics. Keep practicing, and you'll become a pro at linear equations in no time!