Y-intercept Of Parallel Lines: A Step-by-Step Solution
Hey guys! Ever stumbled upon a math problem that seems like a puzzle? Today, we're going to crack one of those – a question involving parallel lines and their y-intercepts. This is a classic problem in coordinate geometry, and understanding it can really boost your problem-solving skills. We'll break it down step-by-step, so you can follow along easily. Let's dive in!
Understanding the Problem
The problem states that we have two lines, line p and line q. Line p passes through the point (1, 2), and it's parallel to line q. Now, line q passes through two points: (-2, 3) and (3, 4). The core question we need to answer is: where does line q intersect the y-axis? In other words, what is the y-intercept of line q?
To tackle this, we'll need to use a few key concepts from coordinate geometry. We need to remember how to find the slope of a line given two points, what it means for lines to be parallel, and how to determine the equation of a line. Don't worry if these sound a bit intimidating – we'll walk through each concept.
Keywords to Focus On
Before we jump into the solution, let's pinpoint the main keywords in the problem. These are our clues! We have "parallel lines," which tells us something about their slopes. We have "y-intercept," which is what we're ultimately trying to find. And we have the coordinates of points on the lines, which are crucial for calculating slopes and equations. Keeping these keywords in mind will help us stay on track.
Why This Problem Matters
So, why are we even bothering with this problem? Well, understanding parallel lines and y-intercepts isn't just about acing a math test. These concepts pop up in various real-world applications. Think about architecture, where parallel lines are fundamental to building design. Or consider navigation, where understanding slopes and intercepts can help you plot a course. Even in computer graphics, these concepts are used to create images and animations.
By mastering this type of problem, you're not just learning math – you're developing critical thinking and problem-solving skills that will be valuable in many areas of life. Now, let's get down to business and solve this puzzle!
Finding the Slope of Line q
The first step in solving this problem is to determine the slope of line q. Remember, the slope of a line tells us how steeply it rises or falls. It's often represented by the letter m. When we have two points on a line, we can calculate the slope using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. In our case, line q passes through the points (-2, 3) and (3, 4). Let's plug these values into our formula:
m = (4 - 3) / (3 - (-2)) m = 1 / 5
So, the slope of line q is 1/5. This means that for every 5 units we move horizontally along the line, we move 1 unit vertically. A positive slope indicates that the line is rising as we move from left to right.
Why Slope is Important
Understanding the slope is crucial because it's a key characteristic of a line. It tells us the line's direction and steepness. In this problem, the slope of line q is not just a number; it's a vital piece of information that will help us find the y-intercept. It also connects line q to line p, since parallel lines have a special relationship with their slopes.
Common Mistakes to Avoid
When calculating the slope, it's easy to make mistakes if you're not careful with the signs or the order of subtraction. Always double-check that you're subtracting the y-coordinates and the x-coordinates in the same order. For example, if you do (y₂ - y₁), make sure you also do (x₂ - x₁), not (x₁ - x₂). Also, remember that a negative slope indicates a line that's falling from left to right.
Now that we've found the slope of line q, we're one step closer to our goal. The next step is to use this information, along with the fact that line p is parallel to line q, to figure out the equation of line q. Let's move on to the next part of the solution!
Finding the Equation of Line q
Now that we know the slope of line q is 1/5, we can move on to finding its equation. There are a couple of ways to do this, but one of the most common is using the point-slope form of a line. This form is particularly handy when you know the slope of a line and a point it passes through. The point-slope form looks like this:
y - y₁ = m(x - x₁)
Where m is the slope, and (x₁, y₁) is a point on the line. We already know the slope of line q (m = 1/5), and we have two points it passes through: (-2, 3) and (3, 4). We can use either of these points in the point-slope form. Let's use the point (-2, 3) for this example.
Plugging in the values, we get:
y - 3 = (1/5)(x - (-2)) y - 3 = (1/5)(x + 2)
Now, to make the equation a bit more familiar, let's convert it to the slope-intercept form, which is y = mx + b, where b is the y-intercept (the value we're trying to find). To do this, we'll distribute the 1/5 and then isolate y:
y - 3 = (1/5)x + 2/5 y = (1/5)x + 2/5 + 3
To add 2/5 and 3, we need a common denominator. We can rewrite 3 as 15/5:
y = (1/5)x + 2/5 + 15/5 y = (1/5)x + 17/5
So, the equation of line q in slope-intercept form is y = (1/5)x + 17/5. This equation tells us everything we need to know about line q, including its y-intercept.
The Power of the Equation
The equation of a line is like a mathematical fingerprint. It uniquely identifies the line and allows us to predict its behavior. In this case, the equation y = (1/5)x + 17/5 tells us the slope of line q (1/5) and its y-intercept (17/5). With this equation, we can find any point on the line, or determine whether a given point lies on the line.
Alternative Approaches
While we used the point-slope form here, you could also solve this problem by plugging the two points (-2, 3) and (3, 4) into the slope-intercept form y = mx + b and solving a system of equations for m and b. This is another valid approach, and it's always good to have multiple tools in your mathematical toolkit.
Now that we have the equation of line q, finding the y-intercept is the final step. Let's see how to do that in the next section!
Finding the Y-Intercept
We've done the hard work of finding the equation of line q: y = (1/5)x + 17/5. Now, the final piece of the puzzle is to identify the y-intercept. Remember, the y-intercept is the point where the line crosses the y-axis. This happens when the x-coordinate is 0.
In the slope-intercept form y = mx + b, the b represents the y-intercept. So, in our equation, y = (1/5)x + 17/5, the y-intercept is simply the constant term, which is 17/5.
To express this as a decimal, we can divide 17 by 5:
17 / 5 = 3.4
So, the y-intercept of line q is 3.4. This means that line q intersects the y-axis at the point (0, 3.4).
Visualizing the Y-Intercept
It's helpful to visualize what we've found. Imagine a coordinate plane with the x and y axes. Line q is a line that slopes upwards gently (since its slope is 1/5). It crosses the y-axis at a point a little above the 3 on the y-axis. That point is (0, 3.4), our y-intercept.
Why the Y-Intercept Matters
The y-intercept is an important characteristic of a line. It tells us where the line starts on the y-axis. In many real-world applications, the y-intercept has a meaningful interpretation. For example, if we were modeling the cost of a service as a function of the number of hours used, the y-intercept might represent the fixed cost, the cost you pay even if you use zero hours.
Checking Our Answer
It's always a good idea to check your answer if you have time. We found that the y-intercept is 3.4. We can plug x = 0 into our equation and see if we get y = 3.4:
y = (1/5)(0) + 17/5 y = 0 + 17/5 y = 17/5 y = 3.4
Yep, it checks out! We've successfully found the y-intercept of line q.
Conclusion
Alright guys, we've done it! We've successfully found the y-intercept of line q by breaking down the problem step-by-step. We started by understanding the problem and identifying the key concepts: parallel lines, slope, and y-intercept. Then, we calculated the slope of line q, found its equation using the point-slope form, and finally, identified the y-intercept from the slope-intercept form.
Key Takeaways
Here are some key takeaways from this problem:
- Parallel lines have the same slope. This is a fundamental property that allows us to connect the information about line p to line q.
- The slope-intercept form (y = mx + b) is a powerful tool for understanding lines. It directly tells us the slope (m) and the y-intercept (b).
- Breaking down a complex problem into smaller steps makes it more manageable. We tackled this problem by finding the slope first, then the equation, and finally the y-intercept.
- Visualizing the problem can help you understand the concepts better. Drawing a quick sketch of the lines and their intercepts can give you a clearer picture.
Keep Practicing!
This type of problem might seem challenging at first, but with practice, you'll become more comfortable with the concepts and the steps involved. Try solving similar problems with different numbers or different scenarios. The more you practice, the more confident you'll become!
So, that's it for this problem. I hope you found this explanation helpful. Remember, math is like a puzzle – it might take some effort to solve, but the feeling of accomplishment when you get the right answer is totally worth it. Keep learning, keep practicing, and I'll see you in the next problem!