Y-Intercept Of Line Through Points P(11, 12) And Q(12, -2)

Hey guys! Let's dive into a common problem in coordinate geometry: finding the y-intercept of a line when we're given two points on that line. In this case, we've got points P = (11, 12) and Q = (12, -2). Our mission, should we choose to accept it, is to determine the y-intercept of the line that gracefully glides through these two points. It might sound intimidating, but trust me, it's totally doable with a few simple steps. So, grab your thinking caps, and let's get started!

Understanding the Basics

Before we jump into the calculations, let's make sure we're all on the same page with the fundamental concepts. First, remember that the y-intercept is the point where the line crosses the y-axis. This happens when the x-coordinate is zero. So, we're looking for a point in the form (0, y), where 'y' is the y-intercept value we need to find. Think of it like this: if you were walking along the line, the y-intercept is where you'd high-five the y-axis!

Next, let's talk about the equation of a line. The most common form we use is the slope-intercept form: y = mx + b. Here, 'm' represents the slope of the line (how steep it is), and 'b' is the y-intercept – exactly what we're trying to find! The slope, m, tells us how much the y-value changes for every one unit change in the x-value. A positive slope means the line goes upwards as you move from left to right, while a negative slope means it goes downwards. A slope of zero means it is a horizontal line.

Another useful formula is the slope formula, which helps us calculate the slope ('m') when we have two points. If we have two points, (x1, y1) and (x2, y2), the slope is given by:

m = (y2 - y1) / (x2 - x1)

This formula essentially calculates the "rise over run" – the change in y divided by the change in x. This gives us a numerical value for how tilted our line is. With these basics in our toolkit, we're ready to tackle the problem head-on!

Step-by-Step Solution

Okay, let’s break down the problem step-by-step to make it super clear. We're given two points, P = (11, 12) and Q = (12, -2), and we need to find the y-intercept of the line that passes through them.

1. Calculate the Slope (m)

The first thing we need to do is find the slope of the line. Remember the slope formula? It’s our best friend here:

m = (y2 - y1) / (x2 - x1)

Let's plug in the coordinates of our points. We can call P (11, 12) as (x1, y1) and Q (12, -2) as (x2, y2). So, we have:

x1 = 11, y1 = 12 x2 = 12, y2 = -2

Now, substitute these values into the slope formula:

m = (-2 - 12) / (12 - 11) m = (-14) / (1) m = -14

So, the slope of our line is -14. That's a pretty steep downward slope! It means that for every one unit we move to the right along the x-axis, the line drops 14 units on the y-axis.

2. Use the Point-Slope Form

Now that we have the slope, we can use the point-slope form of a line equation. This form is super handy when you have a slope and a point (which we do!). The point-slope form looks like this:

y - y1 = m(x - x1)

We already know 'm' (the slope) is -14. We also have two points to choose from, P (11, 12) and Q (12, -2). Let’s use point P (11, 12) for now. So, we have:

x1 = 11, y1 = 12, m = -14

Plug these values into the point-slope form:

y - 12 = -14(x - 11)

3. Convert to Slope-Intercept Form (y = mx + b)

Our next step is to convert the equation from point-slope form to slope-intercept form (y = mx + b). This will make it much easier to identify the y-intercept. To do this, we need to distribute the -14 and then isolate 'y':

y - 12 = -14(x - 11) y - 12 = -14x + 154 (Distribute -14) y = -14x + 154 + 12 (Add 12 to both sides) y = -14x + 166

Voila! We now have the equation of the line in slope-intercept form: y = -14x + 166.

4. Identify the Y-Intercept

Remember, the y-intercept is the 'b' value in the slope-intercept form (y = mx + b). In our equation, y = -14x + 166, the 'b' value is 166. Therefore, the y-intercept is 166. This means the line crosses the y-axis at the point (0, 166).

Alternative Method: Using Point Q

Just to show you that it works no matter which point you use, let's quickly run through the same steps using point Q (12, -2). We already have the slope, m = -14.

1. Use the Point-Slope Form with Point Q

Using the point-slope form y - y1 = m(x - x1) with point Q (12, -2), we get:

y - (-2) = -14(x - 12) y + 2 = -14(x - 12)

2. Convert to Slope-Intercept Form

Distribute the -14 and isolate 'y':

y + 2 = -14x + 168 y = -14x + 168 - 2 (Subtract 2 from both sides) y = -14x + 166

See? We get the same equation, y = -14x + 166, and the same y-intercept, 166. So, it doesn't matter which point you choose; you'll arrive at the same answer.

Conclusion

So, there you have it! The y-intercept of the line passing through the points P = (11, 12) and Q = (12, -2) is 166. We found this by first calculating the slope using the slope formula, then using the point-slope form of a line equation, and finally converting it to slope-intercept form to easily identify the y-intercept. The key takeaway here is that by understanding these fundamental concepts and formulas, you can tackle similar problems with confidence. Whether it's calculating slopes or converting between different forms of line equations, you've got the tools to succeed. Remember to always double-check your calculations and ensure each step aligns with the goal: finding the y-intercept.

Coordinate geometry can seem daunting at first, but breaking it down into smaller, manageable steps, like we did here, makes it much less intimidating. Keep practicing, keep exploring, and you'll become a math whiz in no time. So, next time you encounter a problem like this, you'll be ready to knock it out of the park!