Finding The Equation Of A Perpendicular Line Line K In Slope-Intercept Form

by Scholario Team 76 views

Hey guys! Let's dive into a fun math problem today that involves finding the equation of a line. Specifically, we're going to figure out the equation of a line k that's perpendicular to another line j, and we'll write our final answer in the ever-popular slope-intercept form. Ready? Let's get started!

Understanding the Problem

Okay, so here's the situation. We have line j with the equation y + 6 = 6(x - 1). We also know that line k is special – it's perpendicular to line j. Remember what perpendicular means? It means the two lines intersect at a right angle (90 degrees). And to make things even more interesting, line k passes through the point (8, -4). Our mission, should we choose to accept it, is to find the equation of line k and express it in slope-intercept form, which looks like y = mx + b, where m is the slope and b is the y-intercept.

Before we jump into solving, let's highlight the key pieces of information we've got:

  • Equation of line j: y + 6 = 6(x - 1)
  • Line k is perpendicular to line j
  • Line k passes through the point (8, -4)
  • Goal: Find the equation of line k in slope-intercept form (y = mx + b)

With these details in mind, let's get to work!

Step 1: Finding the Slope of Line j

The first thing we need to do is figure out the slope of line j. Why? Because the slope of a line perpendicular to j is directly related. To find the slope of j, we need to rewrite its equation in slope-intercept form (y = mx + b). Currently, the equation is y + 6 = 6(x - 1). Let's simplify this:

  • Distribute the 6: y + 6 = 6x - 6
  • Subtract 6 from both sides: y = 6x - 12

Now we have the equation of line j in slope-intercept form: y = 6x - 12. See how easy that was? Remember, the slope (m) is the number multiplying x. So, the slope of line j is 6. Let's call this mj = 6. This is a crucial piece of information!

To recap, by converting the equation of line j to slope-intercept form (y = 6x - 12), we've successfully identified its slope as 6. This slope, mj = 6, is the key to unlocking the slope of the perpendicular line k. Stay with me; we're making great progress!

Step 2: Finding the Slope of Line k

Now comes the cool part! We know that line k is perpendicular to line j. There's a neat little rule about perpendicular lines: their slopes are negative reciprocals of each other. What does that mean? It means if the slope of line j is mj, then the slope of line k (mk) is -1/mj. This is a crucial concept when dealing with perpendicular lines. Understanding this relationship makes finding the equation of line k much easier.

We already found that the slope of line j (mj) is 6. So, to find the slope of line k (mk), we take the negative reciprocal of 6:

  • mk = -1/6

That's it! The slope of line k is -1/6. We're one step closer to finding the full equation. See how the negative reciprocal relationship simplifies things? Instead of guessing or trying different slopes, we used this rule to pinpoint the exact slope of the perpendicular line. This is a powerful technique in coordinate geometry!

Step 3: Using Point-Slope Form

We now know the slope of line k (mk = -1/6) and a point it passes through: (8, -4). To write the equation of a line when you have a slope and a point, we use something called the point-slope form. This is another handy tool in our math arsenal. The point-slope form looks like this:

  • y - y1 = m( x - x1 )

Where:

  • m is the slope of the line
  • (x1, y1) is a point on the line

In our case, we know mk = -1/6 and the point (8, -4). So, we can plug these values into the point-slope form:

  • y - (-4) = (-1/6)(x - 8)

Notice how we substituted -4 for y1 and 8 for x1. The double negative on the left side (y - (-4)) will become a plus sign in the next step. This step is all about applying the point-slope form correctly. By substituting the known values, we've created an equation that represents line k. Now, let's simplify this equation to get it into the slope-intercept form we're aiming for.

Step 4: Converting to Slope-Intercept Form

Our final step is to take the equation we got from the point-slope form and transform it into slope-intercept form (y = mx + b). This will give us the final equation of line k in the format we need. Here's our equation from the previous step:

  • y - (-4) = (-1/6)(x - 8)

Let's simplify this step-by-step:

  1. Simplify the left side: y + 4 = (-1/6)(x - 8)
  2. Distribute the -1/6 on the right side: y + 4 = (-1/6)x + 8/6
  3. Simplify 8/6: y + 4 = (-1/6)x + 4/3
  4. Subtract 4 from both sides to isolate y: y = (-1/6)x + 4/3 - 4
  5. To subtract 4, we need a common denominator. Let's rewrite 4 as 12/3: y = (-1/6)x + 4/3 - 12/3
  6. Combine the fractions: y = (-1/6)x - 8/3

And there we have it! The equation of line k in slope-intercept form is:

  • y = (-1/6)x - 8/3*

We've successfully navigated through all the steps, from finding the slope of line j to using the point-slope form and finally converting to slope-intercept form. Great job, guys! This final equation tells us the slope of line k is -1/6 (as we calculated earlier) and its y-intercept is -8/3.

Conclusion

So, to recap, we were given the equation of line j, the fact that line k is perpendicular to j, and a point that line k passes through. Our mission was to find the equation of line k in slope-intercept form. We tackled this challenge by:

  1. Finding the slope of line j by converting its equation to slope-intercept form.
  2. Using the negative reciprocal relationship to determine the slope of line k.
  3. Employing the point-slope form to create an equation for line k.
  4. Converting that equation to slope-intercept form to get our final answer: y = (-1/6)x - 8/3.

This problem highlights several key concepts in coordinate geometry, including slope, perpendicular lines, slope-intercept form, and point-slope form. Mastering these concepts is super useful for tackling more complex math problems down the road. Keep practicing, and you'll become a pro at finding equations of lines in no time! If you guys have any questions feel free to ask!

And that's a wrap! We successfully found the equation of line k. You guys did awesome! Keep up the great work, and remember, math can be fun when you break it down step-by-step. Until next time!