Parallel Line Equation: Step-by-Step Solution
Have you ever wondered how to find the equation of a line that runs perfectly parallel to another, while also passing through a specific point? It might sound a bit tricky at first, but trust me, it's a super useful skill in math and even in real-life situations! In this guide, we're going to break down the process step-by-step, making it easy for anyone to understand. We'll focus on a specific example: finding the equation of a line parallel to y = -4x - 5 and passing through the point (0, 10). So, grab your pencils, and let's dive in!
Understanding Parallel Lines
Before we jump into the equation itself, let's quickly recap what parallel lines actually are. In simple terms, parallel lines are lines that run in the exact same direction. Think of train tracks – they go on and on, never meeting, because they are perfectly parallel. What makes them parallel mathematically? It's all about their slopes. The slope of a line tells us how steep it is – whether it's rising quickly, falling gradually, or perfectly flat. Parallel lines have the same slope. This is the golden rule we'll use to solve our problem. So, when you hear "parallel lines," remember: same slope! This concept is crucial because it simplifies finding the equation of a parallel line significantly. We don't have to reinvent the wheel; we just need to use the existing slope and adjust for the new point. This understanding forms the foundation for all the calculations we'll do next. Keep this in mind as we move forward, and you'll find the process much smoother and easier to grasp.
Identifying the Slope of the Given Line
Okay, now that we know what parallel lines are, let's look at our given equation: y = -4x - 5. This equation is in a special form called slope-intercept form. Slope-intercept form is written as y = mx + b, where m represents the slope of the line, and b represents the y-intercept (the point where the line crosses the y-axis). This form is super handy because it immediately tells us the slope and y-intercept of the line. Looking at our equation, y = -4x - 5, we can directly identify the slope. The number multiplying x is the slope, so in this case, m = -4. This means the line has a negative slope, so it slopes downwards as you move from left to right. A slope of -4 means that for every 1 unit you move to the right along the x-axis, the line goes down 4 units along the y-axis. Now, remember our golden rule: parallel lines have the same slope. So, any line parallel to y = -4x - 5 will also have a slope of -4. This is a crucial piece of information because it gives us the m value for our new equation. We're one step closer to finding the equation of the parallel line!
Using the Point-Slope Form
Alright, we know the slope of our parallel line is -4. Now, we need to make sure it passes through the point (0, 10). To do this, we're going to use another handy equation form called point-slope form. Point-slope form is written as y - y1 = m(x - x1), where m is the slope, and (x1, y1) is a point on the line. This form is perfect for situations where you know the slope and a point, which is exactly what we have! We know m = -4 (the slope we found earlier), and we know our line needs to pass through the point (0, 10), so x1 = 0 and y1 = 10. Now, we just plug these values into the point-slope form: y - 10 = -4(x - 0). See how easy that was? We've now got our equation in point-slope form. This equation represents a line with a slope of -4 that definitely passes through the point (0, 10). However, to make it even clearer and easier to work with, we're going to convert it to slope-intercept form in the next step. So, stick with me, we're almost there!
Converting to Slope-Intercept Form
We've got our equation in point-slope form: y - 10 = -4(x - 0). Now, let's make it even more user-friendly by converting it to slope-intercept form (y = mx + b). This form makes it super easy to visualize the line and identify its y-intercept. To convert, we just need to do a little bit of algebra to isolate y on one side of the equation. First, let's simplify the right side of the equation: -4(x - 0) becomes -4x. So now we have: y - 10 = -4x. Next, to get y by itself, we add 10 to both sides of the equation: y - 10 + 10 = -4x + 10. This simplifies to: y = -4x + 10. Ta-da! We've done it! Our equation is now in slope-intercept form. We can clearly see that the slope (m) is -4 (which we already knew it had to be, since it's parallel to the original line), and the y-intercept (b) is 10. This means the line crosses the y-axis at the point (0, 10), exactly as we wanted. This conversion step is super important because it makes the equation easy to understand and use. We can now quickly graph the line, find other points on the line, and use it for various other mathematical purposes.
The Final Equation
After all our hard work, we've arrived at the final answer! The equation of the line that is parallel to y = -4x - 5 and passes through the point (0, 10) is: y = -4x + 10***. Isn't that satisfying? We started with a seemingly complex problem, broke it down into manageable steps, and solved it! Let's quickly recap what we did. First, we understood the concept of parallel lines and their slopes. We identified the slope of the given line as -4. Then, we used the point-slope form with the slope and the given point (0, 10) to create an equation. Finally, we converted that equation to slope-intercept form to get our final answer. This process might seem like a lot of steps, but with practice, it becomes second nature. And remember, the key is to understand each step and why we're doing it. This not only helps you solve this specific problem but also gives you a solid foundation for tackling other math challenges. So, congratulations! You've successfully found the equation of a parallel line. Now you can confidently apply this knowledge to future problems.
