Finding The Gradient Of A Perpendicular Line

Hey guys! Ever stumbled upon a math problem involving lines, gradients, and perpendicularity? Don't worry, it's a common one, and we're going to break it down together. Let's dive into a problem where we need to find the gradient of a line that's perpendicular to another. This kind of problem often pops up in coordinate geometry, and understanding the concept is super useful. So, grab your pencils, and let's get started!

Understanding the Problem

The problem we're tackling today goes something like this: We're given a line g defined by the equation 2x - y = 29. This line passes through a specific point, (4, -7). We also know that there's another line, h, that's perpendicular to g. Our mission, should we choose to accept it (and we do!), is to find the gradient (or slope) of line h.

Before we jump into the solution, let's make sure we're all on the same page with some key concepts:

• Gradient (or Slope): Think of the gradient as the 'steepness' of a line. It tells us how much the line rises (or falls) for every unit we move to the right. Mathematically, it's represented as the change in y divided by the change in x. We often use the letter 'm' to denote the gradient.
• Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle (90 degrees). There's a special relationship between the gradients of perpendicular lines, which we'll explore shortly.
• Equation of a Line: The equation 2x - y = 29 is a linear equation, and it represents a straight line. We can rewrite it in different forms, such as the slope-intercept form (y = mx + c), which makes it easier to identify the gradient and y-intercept.

With these concepts in mind, let's roll up our sleeves and get to solving the problem!

Step 1: Finding the Gradient of Line g

The first thing we need to do is figure out the gradient of line g. Remember, line g is given by the equation 2x - y = 29. To find the gradient, the easiest way is to rewrite this equation in the slope-intercept form, which is y = mx + c. In this form, 'm' represents the gradient, and 'c' represents the y-intercept (the point where the line crosses the y-axis).

So, let's rearrange the equation 2x - y = 29 to get it into the y = mx + c form:

1. Start with 2x - y = 29
2. Subtract 2x from both sides: -y = -2x + 29
3. Multiply both sides by -1: y = 2x - 29

Now, our equation is in the slope-intercept form (y = mx + c). By comparing this to y = 2x - 29, we can clearly see that the gradient of line g (which we'll call m_g) is 2. That's the coefficient of x! We did it!

So, m_g = 2. Keep this value handy, because we'll need it in the next step.

Step 2: Using the Perpendicularity Condition

Here's where the magic happens! We know that line h is perpendicular to line g. There's a golden rule about the gradients of perpendicular lines: the product of their gradients is -1. This is a crucial concept, so let's say it again for emphasis: If two lines are perpendicular, then m_1 * m_2 = -1, where m_1 and m_2 are their gradients.

In our case, we know the gradient of line g (m_g = 2), and we want to find the gradient of line h (let's call it m_h). Using the perpendicularity condition, we can set up an equation:

m_g * m_h = -1

Substitute the value of m_g (which is 2) into the equation:

2 * m_h = -1

Now, we just need to solve for m_h. To do this, divide both sides of the equation by 2:

m_h = -1 / 2

So, the gradient of line h is -1/2. Awesome! We're one step closer to cracking this problem.

Step 3: The Final Answer

We've done the hard work, guys! We found that the gradient of line h (m_h) is -1/2. Now, let's go back to the original problem and see if we can match our answer to the given options.

The options were:

(A) -3/2 (B) -1/2 (C) 1/2 (D) 1 (E) 3/2

Look at that! Our calculated gradient, -1/2, matches option (B). So, the correct answer is (B).

We nailed it! By understanding the concepts of gradients, perpendicular lines, and the slope-intercept form of a linear equation, we were able to solve this problem step-by-step. High five!

Key Takeaways

Before we wrap up, let's recap the key takeaways from this problem:

• Slope-Intercept Form: Rewriting a linear equation in the form y = mx + c makes it easy to identify the gradient (m) and y-intercept (c).
• Perpendicularity Condition: The product of the gradients of two perpendicular lines is -1. This is a super important rule to remember!
• Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This makes the problem less intimidating and easier to solve.

Let's Practice!

Now that we've conquered this problem together, why not try some similar ones on your own? Practice makes perfect, and the more you work with these concepts, the more confident you'll become. You can try varying the equation of line g or the point it passes through. You can even try problems where you're given the gradient of line h and need to find the gradient of line g.

Here’s a practice question for you:

Line p is defined by the equation 3x + y = 7 and passes through the point (1, 4). Line q is perpendicular to line p. What is the gradient of line q?

Give it a shot, and feel free to share your solution in the comments below! We can learn from each other and help each other grow.

Conclusion

So, guys, that's how you tackle a problem involving gradients and perpendicular lines. Remember the key concepts, follow a step-by-step approach, and don't be afraid to practice! Math can be challenging, but it's also incredibly rewarding when you crack a tough problem. Keep up the awesome work, and I'll see you in the next math adventure! You got this! Remember, understanding the relationship between gradients and perpendicular lines is a powerful tool in your mathematical arsenal. By applying this knowledge, you can solve a wide range of problems in coordinate geometry and beyond. So keep practicing, keep exploring, and keep pushing your mathematical boundaries! You're doing great!