Perpendicular Line Equation: Step-by-Step Solution

Let's dive into how to find the equation of a line that's perpendicular to another line and passes through a specific point. This is a classic problem in coordinate geometry, and we'll break it down step by step so it's super clear. Guys, get ready to sharpen those pencils (or keyboards!).

Understanding the Problem

So, the question we're tackling is this: How do we find the equation of a line, expressed in the slope-intercept form, that is perpendicular to the given line 2x + 12y = -1 and, at the same time, goes through the point (0,9)?

This involves a few key concepts, so let's quickly recap them before we jump into the solution. First, we need to understand slope-intercept form, which is a way of writing linear equations. Second, we need to know what perpendicular lines are and how their slopes relate to each other. Finally, we need to remember how to use a point and a slope to define a line. If you feel rusty on any of these, don't worry! We'll go through them.

Slope-Intercept Form

The slope-intercept form of a line's equation is a super useful way to represent linear equations. It looks like this: y = mx + b. In this equation:

• y is the dependent variable (usually plotted on the vertical axis).
• x is the independent variable (usually plotted on the horizontal axis).
• m is the slope of the line, which tells us how steep the line is and in what direction it's going. It's the change in y for every unit change in x.
• b is the y-intercept, which is the point where the line crosses the y-axis. It's the value of y when x is 0.

This form is awesome because it immediately tells you two important things about the line: its slope and where it crosses the y-axis. This makes it easy to visualize the line and graph it.

Perpendicular Lines

Perpendicular lines are lines that intersect each other at a right angle (90 degrees). The relationship between their slopes is key to solving our problem. If you have two lines that are perpendicular, the product of their slopes is always -1. In other words, if one line has a slope of m1 and the other has a slope of m2, then for them to be perpendicular:

m1 * m2 = -1

Another way to think about this is that the slope of a line perpendicular to another is the negative reciprocal of the original slope. So, if a line has a slope of, say, 2, the slope of a line perpendicular to it would be -1/2.

Using a Point and Slope to Define a Line

To define a unique line, you need two pieces of information. One way to do this is with a point the line passes through and its slope. There are a couple of ways to use this information to get the equation of the line. One common method is to use the point-slope form, which we'll see later. Another is to directly plug the point's coordinates into the slope-intercept form and solve for the y-intercept.

Solving the Problem: Step-by-Step

Alright, with those concepts in mind, let's break down how to solve the problem step-by-step.

Step 1: Find the Slope of the Given Line

The given line has the equation 2x + 12y = -1. To find its slope, we need to rearrange this equation into slope-intercept form (y = mx + b). Let's do that:

1. Subtract 2x from both sides: 12y = -2x - 1
2. Divide both sides by 12: y = (-2/12)x - 1/12
3. Simplify the fraction: y = (-1/6)x - 1/12

Now we can clearly see that the slope of the given line is -1/6. This is our m1.

Step 2: Find the Slope of the Perpendicular Line

Remember, the slope of a line perpendicular to the given line is the negative reciprocal of its slope. So, to find the slope (m2) of the perpendicular line, we do the following:

1. Take the reciprocal of -1/6, which is -6/1 or -6.
2. Change the sign, making it positive 6.

So, the slope of the perpendicular line is 6. This is our m2.

Step 3: Use the Point-Slope Form (Optional) or Slope-Intercept Form

We know the perpendicular line has a slope of 6 and passes through the point (0, 9). We can use this information to find the equation of the line in a couple of ways.

Method 1: Using Slope-Intercept Form Directly

Since we're aiming for slope-intercept form (y = mx + b), and we already know the slope (m = 6), we have:

y = 6x + b

Now, we just need to find the y-intercept (b). We know the line passes through (0, 9). Notice that the x-coordinate is 0, meaning this point is the y-intercept! So, b = 9.

Therefore, the equation of the perpendicular line is:

y = 6x + 9

Method 2: Using Point-Slope Form

The point-slope form of a linear equation is:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is a point on the line. We know m = 6 and the line passes through (0, 9), so we have:

y - 9 = 6(x - 0)

Now, let's simplify and convert to slope-intercept form:

y - 9 = 6x

Add 9 to both sides:

y = 6x + 9

See? We get the same answer!

Step 4: State the Answer

The equation of the line that is perpendicular to 2x + 12y = -1 and passes through the point (0, 9) is:

y = 6x + 9

Common Mistakes to Avoid

• Forgetting to take the negative reciprocal: A common mistake is to find the reciprocal of the slope but forget to change the sign. Remember, perpendicular lines have slopes that are negative reciprocals of each other.
• Not rearranging the original equation: You must convert the given equation to slope-intercept form before identifying the slope.
• Incorrectly applying the point-slope form: Double-check that you're plugging the coordinates into the correct places in the formula.

Practice Makes Perfect

Finding equations of lines, especially perpendicular ones, is a fundamental skill in algebra and geometry. The best way to master it is to practice! Try working through similar problems with different given lines and points. You'll get the hang of it in no time.

Conclusion

So, there you have it! We've successfully found the equation of a line perpendicular to a given line and passing through a specified point. We revisited the concepts of slope-intercept form and the relationship between the slopes of perpendicular lines. Remember, the key is to break the problem down into manageable steps and use the right formulas. Keep practicing, and you'll be a pro at this in no time. Nice job, guys!