Finding The Y-Intercept Of A Perpendicular Line A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving deep into a classic problem in coordinate geometry: finding the y-intercept of a line that's perpendicular to another line and passes through a specific point. This might sound like a mouthful, but don't worry, we'll break it down step-by-step, making sure everyone, from math newbies to seasoned pros, can follow along. We'll not only solve this specific problem but also equip you with the knowledge and skills to tackle similar challenges with confidence. So, grab your pencils, notebooks, and let's get started on this exciting mathematical journey!
Understanding the Fundamentals
Before we jump into the problem itself, let's brush up on some key concepts that will be crucial for our success. These include the slope-intercept form of a line, the relationship between slopes of perpendicular lines, and how to use the point-slope form to determine the equation of a line.
Slope-Intercept Form: The Line's Identity Card
The slope-intercept form is a way of writing the equation of a line that immediately tells us two important things about the line: its slope and its y-intercept. The general form looks like this:
y = mx + b
Where:
- m represents the slope of the line. 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, a slope of zero means the line is horizontal, and an undefined slope means the line is vertical.
- b represents the y-intercept. The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is equal to 0. This is a crucial point, and we will find this in our problem.
Understanding the slope-intercept form is like having a secret decoder for lines. It allows us to quickly visualize and analyze the behavior of a line just by looking at its equation.
Perpendicular Lines: A Meeting at Right Angles
Perpendicular lines are lines that intersect at a right angle (90 degrees). This special relationship between lines leads to a very important connection between their slopes. If two lines are perpendicular, the product of their slopes is always -1. In other words, the slope of one line is the negative reciprocal of the slope of the other line.
Let's say we have a line with a slope of m1. If another line is perpendicular to it, its slope (m2) will be:
m2 = -1 / m1
This relationship is essential for solving problems involving perpendicular lines. It allows us to find the slope of a line if we know the slope of a line perpendicular to it, and vice versa. In our problem, this property will be key to finding the slope of the line we're interested in.
Point-Slope Form: Building a Line from a Point and a Slope
The point-slope form is another way to represent the equation of a line. It's particularly useful when we know a point that the line passes through and the slope of the line. The general form looks like this:
y - y1 = m(x - x1)
Where:
- m is the slope of the line.
- (x1, y1) is a point that the line passes through.
The point-slope form allows us to construct the equation of a line using minimal information. Once we have the equation in point-slope form, we can easily convert it to slope-intercept form by simplifying and isolating y. This is exactly what we'll do in our problem to find the equation of the perpendicular line.
Tackling the Problem: A Step-by-Step Solution
Now that we've reviewed the fundamental concepts, let's get our hands dirty and solve the problem at hand. Our goal is to find the y-intercept of the line perpendicular to the line y = -3/4x + 5 that includes the point (-3, -3). Let's break this down into manageable steps.
Step 1: Identify the Slope of the Given Line
The given line is in slope-intercept form: y = -3/4x + 5. By comparing this to the general form y = mx + b, we can easily identify the slope (m) of this line. In this case, the slope is -3/4.
m1 = -3/4
This is our starting point. We know the slope of the line we're given, and we'll use this to find the slope of the perpendicular line.
Step 2: Determine the Slope of the Perpendicular Line
As we discussed earlier, the slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, to find the slope of the perpendicular line (m2), we take the negative reciprocal of -3/4.
m2 = -1 / (-3/4) = 4/3
So, the slope of the line perpendicular to y = -3/4x + 5 is 4/3. We're halfway there! We now know the slope of the line we're trying to find.
Step 3: Use the Point-Slope Form to Find the Equation of the Perpendicular Line
We know the slope of the perpendicular line (4/3) and a point it passes through (-3, -3). This is the perfect scenario for using the point-slope form:
y - y1 = m(x - x1)
Plug in the values we know:
y - (-3) = (4/3)(x - (-3))
Simplify the equation:
y + 3 = (4/3)(x + 3)
This is the equation of the perpendicular line in point-slope form. But we're not quite done yet. We need to find the y-intercept, so we need to convert this to slope-intercept form.
Step 4: Convert to Slope-Intercept Form
To convert the equation to slope-intercept form (y = mx + b), we need to distribute and isolate y. Let's start by distributing the 4/3 on the right side:
y + 3 = (4/3)x + 4
Now, subtract 3 from both sides to isolate y:
y = (4/3)x + 4 - 3
y = (4/3)x + 1
We've done it! The equation of the perpendicular line in slope-intercept form is y = (4/3)x + 1.
Step 5: Identify the Y-Intercept
Now that we have the equation in slope-intercept form, finding the y-intercept is a breeze. Remember, the y-intercept is the b value in the equation y = mx + b. In our case, b is 1.
Therefore, the y-intercept of the line perpendicular to y = -3/4x + 5 that includes the point (-3, -3) is 1.
Key Takeaways and General Strategies
We've successfully solved the problem, but let's pause for a moment to reflect on the key takeaways and general strategies we can apply to similar problems in the future.
- Master the Forms: Understanding the slope-intercept form (y = mx + b) and the point-slope form (y - y1 = m(x - x1)) is crucial. These forms are your tools for working with linear equations.
- Perpendicular Slopes are Key: Remember the relationship between the slopes of perpendicular lines. The slope of one line is the negative reciprocal of the slope of the other.
- Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This makes the process less daunting and reduces the chances of making errors.
- Visualize: Whenever possible, try to visualize the problem. Sketching a graph can often provide valuable insights and help you understand the relationships between the lines and points.
Practice Makes Perfect: Try These Problems
To solidify your understanding, try solving these similar problems:
- What is the y-intercept of the line perpendicular to the line y = 2x - 3 that includes the point (2, 1)?
- Find the y-intercept of the line perpendicular to the line y = -1/2x + 4 that passes through the point (-4, 0).
- Determine the y-intercept of the line perpendicular to the line y = 5x + 1 that includes the point (0, -2).
Working through these practice problems will help you build confidence and develop your problem-solving skills. Remember, mathematics is a skill that improves with practice.
Conclusion: You've Conquered Perpendicular Lines!
Congratulations! You've successfully navigated the world of perpendicular lines and y-intercepts. We've not only solved a specific problem but also explored the underlying concepts and strategies that will help you tackle a wide range of similar challenges. Remember to practice, stay curious, and keep exploring the fascinating world of mathematics. Keep up the great work, guys! You've got this!