Perpendicular Line Equation: Slope-Intercept Form Explained
Hey guys! Today, we're diving into a classic math problem: finding the equation of a line that's perpendicular to another line and passes through a specific point. Specifically, we'll tackle this question: How do we determine the slope-intercept equation of a line that is perpendicular to the line y-4 = -2/3(x-6) and passes through the point (-2, -2)? This might sound a bit complex at first, but don't worry! We'll break it down step-by-step, making sure you understand the logic behind each move. So, grab your pencils, and let's get started!
Understanding Slope-Intercept Form and Perpendicular Lines
Before we jump into solving the problem directly, let's quickly recap two key concepts: the slope-intercept form of a linear equation and the relationship between slopes of perpendicular lines. These are the building blocks for our solution, and having a solid grasp of them is super important.
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations that makes it easy to identify the slope and y-intercept of the line. The general form is:
y = mx + b
Where:
- m represents the slope of the line.
- b represents the y-intercept (the point where the line crosses the y-axis).
The slope tells us how steep the line is and whether it's increasing or decreasing as we move from left to right. A positive slope means the line goes upwards, a negative slope means it goes downwards, and a slope of zero means it's a horizontal line. The y-intercept is simply the y-coordinate of the point where the line intersects the y-axis. Knowing these two values gives us a complete picture of the line's position and orientation on the coordinate plane.
Perpendicular Lines
Now, let's talk about perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). There's a special relationship between the slopes of perpendicular lines: they are negative reciprocals of each other. This means if one line has a slope of m, a line perpendicular to it will have a slope of -1/m.
Think about it this way: if a line has a steep positive slope, a line perpendicular to it will have a shallow negative slope, and vice versa. This inverse relationship ensures they meet at a perfect right angle. This concept is crucial for solving our problem, so make sure you're comfortable with it. Remember, to find the perpendicular slope, you flip the fraction and change the sign!
Step-by-Step Solution
Okay, with those concepts fresh in our minds, let's tackle the problem step-by-step. We need to find the equation of a line that's perpendicular to y - 4 = -2/3(x - 6) and passes through the point (-2, -2). Here's how we'll do it:
Step 1: Find the Slope of the Given Line
First, we need to determine the slope of the line given by the equation y - 4 = -2/3(x - 6). To do this, we'll rewrite the equation in slope-intercept form (y = mx + b). Let's distribute the -2/3 on the right side:
y - 4 = -2/3x + 4
Now, add 4 to both sides to isolate y:
y = -2/3x + 8
Great! Now the equation is in slope-intercept form. We can clearly see that the slope of the given line, which we'll call m1, is -2/3. Identifying the slope of the original line is the crucial first step, as it allows us to determine the slope of the perpendicular line we're trying to find.
Step 2: Calculate the Slope of the Perpendicular Line
Remember, the slope of a line perpendicular to another is the negative reciprocal. So, to find the slope of our perpendicular line (m2), we need to flip the fraction and change the sign of m1.
m1 = -2/3
Flipping the fraction gives us 3/2, and changing the sign makes it positive. Therefore:
m2 = 3/2
So, the slope of the line we're looking for is 3/2. This means for every 2 units we move to the right on the graph, the line will move 3 units upwards. This positive slope tells us the line will be increasing as we move from left to right.
Step 3: Use the Point-Slope Form
Now that we have the slope of the perpendicular line (m2 = 3/2) and a point it passes through (-2, -2), we can use the point-slope form of a linear equation to find its equation. The point-slope form is:
y - y1 = m(x - x1)
Where:
- m is the slope.
- (x1, y1) is the given point.
Plug in our values: m = 3/2 and (x1, y1) = (-2, -2):
y - (-2) = 3/2(x - (-2))
Simplify:
y + 2 = 3/2(x + 2)
The point-slope form is a powerful tool because it allows us to directly incorporate the slope and a known point on the line. This form is often an intermediate step towards finding the slope-intercept form, which is our ultimate goal.
Step 4: Convert to Slope-Intercept Form
Our final step is to convert the equation from point-slope form to slope-intercept form (y = mx + b). To do this, we need to distribute the 3/2 on the right side and then isolate y.
Starting with:
y + 2 = 3/2(x + 2)
Distribute the 3/2:
y + 2 = 3/2x + 3
Subtract 2 from both sides to isolate y:
y = 3/2x + 1
And there we have it! The equation of the line perpendicular to y - 4 = -2/3(x - 6) and passing through the point (-2, -2), in slope-intercept form, is:
y = 3/2x + 1
Conclusion
So, we've successfully navigated the steps to find the equation of a perpendicular line in slope-intercept form. We started by understanding the basic concepts of slope-intercept form and the relationship between slopes of perpendicular lines. Then, we systematically worked through the problem: finding the slope of the given line, calculating the perpendicular slope, using the point-slope form, and finally, converting to slope-intercept form.
Remember, the key to mastering these types of problems is practice. Work through similar examples, and don't be afraid to break down the problem into smaller, more manageable steps. With a little effort, you'll be solving these equations like a pro! And if you ever get stuck, remember the relationship between perpendicular slopes: flip the fraction and change the sign! Guys, keep practicing, and you'll get there. Good luck, and happy solving!