Finding The Equation Of A Line Passing Through A Point At An Angle

Hey guys! Let's dive into a cool math problem today. We're going to figure out the equation of a line that goes through a specific point and makes a particular angle with the x-axis. Sounds interesting, right? So, buckle up, and let’s get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We need to find the equation of a line. Remember, a line's equation usually looks like this: y = mx + c. Here, m is the slope (how steep the line is), and c is the y-intercept (where the line crosses the y-axis). We know two crucial things about this line: it passes through the point (-3, 5), and it makes an angle of 53.13010235° with the x-axis. This angle information is super important because it's directly related to the slope of the line. Think about it – the steeper the angle, the steeper the line, and therefore, the bigger the slope. So, our mission is to use these two pieces of information to find m and c, and then we'll have our equation!

To really grasp this, imagine a coordinate plane. Plot the point (-3, 5). Now, visualize a line going through that point. This line isn't just any line; it’s tilted at a specific angle relative to the horizontal x-axis. The angle of inclination tells us the line's direction and steepness. When we say the angle is 53.13010235°, we're talking about the angle formed between the line and the positive x-axis, measured counterclockwise. This angle is our key to unlocking the slope. The point (-3, 5) gives us a fixed location the line must pass through. Combining the angle and the point, we can pinpoint the exact line we're looking for. We will need to leverage the relationship between the angle of inclination and the slope. The slope, often denoted as 'm', is the tangent of the angle of inclination. Once we calculate the slope, we can use the point-slope form of a line equation to find the equation in the desired y = mx + c format. This problem beautifully combines geometry (angles and points) with algebra (linear equations), giving us a practical application of these mathematical concepts. Understanding these fundamentals is crucial for tackling more complex problems in coordinate geometry and calculus. So, let's dive into the solution and see how it all comes together!

Finding the Slope (m)

Okay, so how do we use that angle to find the slope? Here's the magic trick: the slope (m) is equal to the tangent of the angle. Remember trigonometry? The tangent function (tan) relates an angle to the ratio of the opposite side to the adjacent side in a right triangle. In our case, the angle is 53.13010235°. So, we need to find the tangent of this angle. You can use a calculator for this, just make sure it's in degree mode! The tangent of 53.13010235° is approximately 1.333. But wait, 1.333 looks a lot like a fraction we know, right? It's very close to 4/3. In fact, for problems like these, it's often nice to express the slope as a fraction because it gives us a clearer picture of the line's rise over run. So, we can say that m = 4/3. This means for every 3 units we move to the right along the x-axis, the line goes up 4 units on the y-axis. That’s a pretty steep line!

The slope, being the tangent of the angle of inclination, tells us the steepness and direction of the line. A positive slope, like the 4/3 we found, means the line slopes upwards from left to right. A larger slope value indicates a steeper incline. Conversely, a negative slope would mean the line slopes downwards. A zero slope represents a horizontal line, and an undefined slope signifies a vertical line. Understanding this connection between the angle and the slope is fundamental in coordinate geometry. It allows us to translate geometric information (the angle) into algebraic terms (the slope), which we can then use to define the line's equation. Now that we've calculated the slope, we are halfway to finding the line's equation. The next step is to use the point (-3, 5) that the line passes through, along with the slope we just found, to determine the y-intercept 'c'. This will complete our equation in the form y = mx + c. So, let's move on to the next step and see how we can use this information to solve for 'c'. Keep up the great work, guys; we're getting closer to the solution!

Finding the Y-Intercept (c)

Now that we know the slope (m = 4/3), we're one step closer to finding the equation. We still need to find the y-intercept (c). Remember, the y-intercept is the point where the line crosses the y-axis. To find c, we can use the point-slope form of a line, or we can simply plug the point (-3, 5) and the slope (4/3) into our equation y = mx + c. Let's do that! We have 5 = (4/3) * (-3) + c. Now it’s just a matter of solving for c. First, let's simplify (4/3) * (-3). That's -4. So, our equation becomes 5 = -4 + c. To isolate c, we add 4 to both sides of the equation: 5 + 4 = c, which means c = 9. So, the y-intercept is 9. This means the line crosses the y-axis at the point (0, 9).

Finding the y-intercept is crucial because it completes the description of our line. The y-intercept gives us a fixed point on the line, and combined with the slope, it uniquely defines the line's position in the coordinate plane. There are alternative methods to find the y-intercept, such as using the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Plugging in our values, we get y - 5 = (4/3)(x - (-3)), which simplifies to y - 5 = (4/3)(x + 3). Expanding and rearranging this equation will also eventually lead us to the same y = mx + c form and the same value for 'c'. This demonstrates the interconnectedness of different forms of linear equations and how they all represent the same underlying line. Now that we've successfully found both the slope (m) and the y-intercept (c), we have all the pieces needed to write the complete equation of the line. This is the final step, and it's where we put everything together. So, let's go ahead and write out the equation and celebrate our solution!

The Final Equation

Alright, we've done the hard work, guys! We found that the slope m is 4/3, and the y-intercept c is 9. Now, we just plug these values back into the equation y = mx + c. So, the equation of the line is y = (4/3)x + 9. And that’s it! We've successfully found the equation of the line that passes through the point (-3, 5) and makes an angle of 53.13010235° with the x-axis. High five!

Let’s quickly recap what we did. We started with a line that had a specific angle to the x-axis and passed through a given point. We used the angle to find the slope by calculating the tangent of the angle. Then, we used the point and the slope to find the y-intercept by plugging these values into the slope-intercept form of a linear equation. Finally, we combined the slope and the y-intercept to write the complete equation of the line. This process is a classic example of how we can use algebra and trigonometry to solve geometric problems. It demonstrates the powerful connection between different areas of mathematics and how they can be used together to understand and describe the world around us. Understanding how to find the equation of a line is a fundamental skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics. So, mastering this concept is a significant step in your mathematical journey. I hope you found this explanation clear and helpful. Keep practicing, and you'll become a pro at solving these types of problems!

Conclusion

So, there you have it! We successfully found the equation of the line: y = (4/3)x + 9. This problem was a great exercise in combining geometry and algebra. Remember, the key is to break down the problem into smaller steps, understand the relationships between the concepts (like angle and slope), and then use the tools you have (like the slope-intercept form) to solve for the unknowns. Keep practicing, and you’ll be solving these problems in your sleep! And that’s a wrap, folks! I hope you enjoyed this math adventure. Until next time, keep those brains buzzing!