Parallel & Perpendicular Lines: Math Problem Solved!

Hey guys! Let's dive into a cool math problem involving parallel and perpendicular lines. We'll be working through a classic geometry scenario where we have a line, and we need to find the equation of a parallel line and the gradient of a perpendicular one. This is a super important concept in understanding linear equations, so let's break it down step-by-step. Get ready to flex those math muscles! We'll explore how to find the equation of a parallel line and determine the gradient of a perpendicular line, making this seemingly complex problem a piece of cake. This type of problem often appears in algebra and geometry, so mastering these concepts will definitely boost your mathematical prowess. So, let's get started and make this problem as clear as possible!

Understanding the Basics: Parallel Lines

Parallel lines, by definition, are lines that never intersect. Think of train tracks – they run alongside each other forever, right? That’s the essence of parallel lines! A super important thing to remember is that parallel lines have the same gradient (slope). The gradient tells us how steeply a line slopes upwards or downwards. If two lines have the same gradient, they'll rise or fall at the same rate, ensuring they never cross paths. This is the cornerstone of solving the first part of our problem. The fact that the lines never meet is the key characteristic here. For example, if you have two parallel lines, and one has a gradient of 3, the other must also have a gradient of 3. Understanding this principle is crucial, as it provides us with the first key element for solving our problem. So, when identifying parallel lines, you will always look for the same gradient. This fundamental concept is absolutely critical for understanding and solving these types of problems. The gradient is the same, no matter what! Remember, the gradient, or slope, is a measure of how quickly the line rises or falls. So, if two lines have the same gradient, they rise or fall at the same rate. This means they will always remain the same distance apart.

The Given Equation

We're given the equation of a line: y = 2x + 4. In this equation, the '2' is the gradient (often represented by the letter 'm'). This tells us the line slopes upwards, and for every one unit we move to the right, the line goes up two units. The '+ 4' is the y-intercept, where the line crosses the y-axis (the vertical line). It's the point (0, 4) on the graph. This y = 2x + 4 equation forms the basis of our whole problem. From this, we know the gradient of the original line. Remember this, as it is key to the first part of the problem. This equation provides us with all the info to find the equation of a parallel line. The gradient is 2, the y-intercept is 4. These are two critical pieces of information. Don't worry if this seems complicated at first; we will break down each step. This equation is the heart of the matter, providing the critical foundation.

Line M and Parallelism

We know that line M is parallel to the line y = 2x + 4. Therefore, line M must also have a gradient of 2. They have the same slope! If the original line has a gradient of 2, the parallel line must also have a gradient of 2. Keep this in mind, as it is the foundation of the question. Because it's parallel, line M also has the gradient of 2. We now know the gradient of line M. Parallel lines share the same slope, and this property is essential to finding the equation of the parallel line, M. This fundamental concept is central to the problem. If line M is parallel, it must share the same gradient.

Determining the Equation of Line M

Okay, now that we've established the gradient of line M (which is 2), we also know that line M passes through point R (3, 5). We have the gradient and a point. Perfect! Now we can determine the equation of line M. Here, we'll use the point-slope form of a linear equation, which is super handy in this situation.

Point-Slope Form

The point-slope form is: y - y₁ = m(x - x₁), where:*

• 'm' is the gradient
• (x₁, y₁) is the point the line passes through

Plugging in the Values

Let’s plug in the values we know: m = 2, x₁ = 3, and y₁ = 5. Our equation becomes: y - 5 = 2(x - 3).

Simplifying the Equation

Now, let's simplify this equation to get it into the more familiar slope-intercept form (y = mx + b).

1. Distribute the 2: y - 5 = 2x - 6
2. Add 5 to both sides: y = 2x - 1

And there you have it! The equation of line M is y = 2x - 1. This is your final answer for part (a). This is the equation of the line that is parallel to the original line and passes through point R (3, 5). By using the point-slope form, we have effectively solved this problem.

Perpendicular Lines: Understanding the Concept

Alright, let's switch gears and talk about perpendicular lines. These are lines that intersect at a right angle (90 degrees). Now, unlike parallel lines, perpendicular lines don’t have the same gradient. Instead, their gradients are negative reciprocals of each other. This means you flip the sign (positive to negative or vice versa) and flip the fraction. Remember this! Perpendicular lines form a perfect right angle. This relationship is a fundamental concept in geometry, essential for solving this problem. This is a critical concept, so let's break it down further. This concept is the key to understanding how to find the gradient of a perpendicular line.

Negative Reciprocal

To find the negative reciprocal of a number, first, you change its sign. Then, if the number is a whole number (like 2), you can write it as a fraction over 1 (2/1), and you flip it (1/2). So, the negative reciprocal of 2 is -1/2. If the gradient of a line is 2, the gradient of a line perpendicular to it is -1/2. Always remember to flip the sign and the fraction. This is how you calculate the negative reciprocal, and this is the gradient. By understanding negative reciprocals, you’re well on your way to solving this part of the problem.

Determining the Gradient of the Perpendicular Line

We're asked to find the gradient of a line perpendicular to line M. We've already determined that the gradient of line M is 2 (from the equation y = 2x - 1). Now, let’s find the gradient of a line perpendicular to M.

Finding the Negative Reciprocal

1. Start with the gradient of line M, which is 2.
2. Change the sign from positive to negative: -2
3. Write it as a fraction: -2/1
4. Flip the fraction: -1/2

So, the gradient of the line perpendicular to line M is -1/2. That’s the answer to part (b). This gradient tells us about the steepness and direction of the perpendicular line. This -1/2 is the slope of the line perpendicular to line M. Understanding how to find this is very important. With this knowledge, you can solve many geometry problems. This is the last part of this problem. Well done!

Summary

Here’s a quick recap of what we covered, guys:

• Parallel lines have the same gradient.
• To find the equation of a parallel line, use the gradient and a point on the line in the point-slope form.
• Perpendicular lines have gradients that are negative reciprocals of each other.
• To find the gradient of a perpendicular line, find the negative reciprocal of the original line's gradient.

And that's it! You've successfully navigated a math problem involving parallel and perpendicular lines. You've found the equation of a parallel line and the gradient of a perpendicular one. Keep practicing, and these concepts will become second nature! Feel free to go over this again until you feel completely confident with the material. Keep up the excellent work, and always remember the importance of understanding these fundamentals.