Line Equation: 135° Angle Through (3,4)

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Alright, guys, let's dive into a cool math problem! We're going to figure out how to find the equation of a line. But not just any line – this one makes a 135-degree angle and cruises right through the point (3, 4). Sounds like fun? Let's get started!

Understanding the Basics

Before we jump into solving, let's make sure we're all on the same page with some basics. First, what's an equation of a line? The most common form is the slope-intercept form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The slope tells us how steep the line is, and the y-intercept is where the line crosses the y-axis.

Now, what about angles? Remember that the slope of a line is related to the angle it makes with the x-axis. Specifically, the slope 'm' is the tangent of the angle (θ). So, m = tan(θ). This is super important because it links the angle we're given (135 degrees) to the slope we need for our line equation.

Lastly, we need to understand how a point on the line helps us. If we know a point (x, y) that the line passes through, we can plug those values into our equation y = mx + b, along with the slope 'm', to solve for the y-intercept 'b'. This is the final piece of the puzzle that lets us write the complete equation of the line. Got it? Great, let's move on!

Finding the Slope

Okay, so the first thing we need to do is figure out the slope of the line. We know the line makes a 135-degree angle with the x-axis. As we just discussed, the slope 'm' is the tangent of this angle. So, we need to find tan(135°). If you remember your trig, 135 degrees is in the second quadrant, where tangent is negative.

Specifically, tan(135°) = -1. Therefore, the slope of our line, m = -1. Easy peasy, right? Now that we have the slope, we're one step closer to finding the full equation of the line. This is a crucial step, so make sure you're comfortable with using the angle to find the slope. If you're ever unsure, sketch a quick unit circle to visualize the angle and its tangent. Visual aids can be a lifesaver in trigonometry!

Finding the Y-Intercept

Alright, now that we know the slope (m = -1), we can plug it into our slope-intercept equation: y = mx + b. This gives us y = -1x + b, or simply y = -x + b. But we still need to find 'b', the y-intercept.

This is where the point (3, 4) comes in handy. Since the line passes through this point, we know that when x = 3, y = 4. So, we can substitute these values into our equation: 4 = -3 + b. Now we just need to solve for 'b'.

Adding 3 to both sides of the equation, we get b = 4 + 3 = 7. So, the y-intercept is 7. We're almost there! We've found the slope and the y-intercept. Now we just need to put it all together to get the equation of the line.

Writing the Equation

Okay, we've got all the pieces we need. We know the slope m = -1, and we know the y-intercept b = 7. So, we can plug these values back into the slope-intercept form of the equation of a line:

y = mx + b

y = -1x + 7

Or, more simply:

y = -x + 7

And that's it! We've found the equation of the line that forms a 135-degree angle and passes through the point (3, 4). The equation is y = -x + 7. Wasn't that fun?

Alternative Forms of the Equation

While y = -x + 7 is a perfectly valid and common way to represent the equation of the line (slope-intercept form), there are other forms you might encounter or find useful. Let's briefly touch on two of them: the point-slope form and the standard form.

Point-Slope Form

The point-slope form is particularly handy when you know a point on the line (x₁, y₁) and the slope 'm'. It's given by:

y - y₁ = m(x - x₁)

In our case, we know the point (3, 4) and the slope m = -1. Plugging these values in, we get:

y - 4 = -1(x - 3)

This is also a valid equation for the same line. If you simplify this equation, you'll find that it's equivalent to our slope-intercept form.

Standard Form

The standard form of a linear equation is given by:

Ax + By = C

Where A, B, and C are constants, and A and B are not both zero. To convert our slope-intercept form y = -x + 7 to standard form, we can add 'x' to both sides:

x + y = 7

This is the standard form of the equation for our line. Different forms are useful in different contexts, so it's good to be familiar with them.

Visualizing the Line

To really understand what we've found, it's helpful to visualize the line. Imagine a graph with the x and y axes. Our line has a negative slope of -1, which means it goes downwards as you move from left to right. It crosses the y-axis at the point (0, 7). And it passes right through the point (3, 4), as we were told.

If you were to draw this line, you'd see that it indeed forms a 135-degree angle with the x-axis. This visual confirmation can be really helpful in solidifying your understanding of the problem and the solution. There are many online graphing tools you can use to plot the equation y = -x + 7 and see it for yourself. It's a great way to double-check your work and make sure everything makes sense.

Common Mistakes to Avoid

When solving problems like this, there are a few common mistakes that students often make. Here are some things to watch out for:

  • Incorrectly Calculating the Slope: Make sure you know your trigonometry and how to find the tangent of angles, especially those in different quadrants. A mistake here will throw off your entire solution.
  • Forgetting the Negative Sign: Remember that tan(135°) is negative. Don't forget the negative sign when determining the slope.
  • Incorrect Substitution: When plugging in the point (3, 4) into the equation, make sure you substitute the values correctly. It's easy to mix up x and y.
  • Algebra Errors: Be careful with your algebra when solving for 'b'. A simple arithmetic mistake can lead to the wrong answer.
  • Not Checking Your Work: Always double-check your work, especially if you have time. Plug your final equation back into the original problem to make sure it satisfies the given conditions.

By avoiding these common mistakes, you'll be well on your way to solving these types of problems accurately and confidently.

Conclusion

So, there you have it! We successfully found the equation of the line that forms a 135-degree angle and passes through the point (3, 4). We started by understanding the basics of linear equations and slopes, then used trigonometry to find the slope, and finally used the given point to find the y-intercept. We put it all together to get the equation y = -x + 7. We also explored alternative forms of the equation and discussed common mistakes to avoid.

I hope this explanation was helpful and clear. Remember, practice makes perfect, so keep working on these types of problems to improve your skills. And don't be afraid to ask for help if you get stuck. Happy solving, guys!