Finding Equations Of Parallel Lines: A Comprehensive Guide

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Hey everyone! Let's dive into a cool math problem today. We're going to figure out the equations of lines that not only pass through a specific point, which is (4,3), but also run parallel to a given line, 2x - 2y = 11. This is a classic problem that combines a few key concepts in coordinate geometry, so let's break it down step by step.

Understanding Parallel Lines and Their Slopes

First things first, let's recap what it means for lines to be parallel. Parallel lines, my friends, are lines that run in the same direction and never intersect. The most important thing to remember here is that parallel lines have the same slope. The slope, often denoted as 'm', tells us how steep a line is. It's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. So, if we can figure out the slope of the given line, 2x - 2y = 11, we'll know the slope of any line parallel to it. How do we find this slope, you ask? Great question! We need to get the equation into what's called slope-intercept form. This form is your best friend in these situations because it makes the slope super easy to spot. The slope-intercept form looks like this: y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). Let's transform our given equation, 2x - 2y = 11, into this form. We'll start by isolating the 'y' term. Subtract 2x from both sides of the equation, and we get -2y = -2x + 11. Next, we'll divide both sides by -2 to solve for 'y'. This gives us y = x - 11/2. Now, look at that! Our equation is in slope-intercept form, and we can clearly see that the slope, 'm', is 1 (since the coefficient of x is 1). So, any line parallel to 2x - 2y = 11 will also have a slope of 1. This is a crucial piece of the puzzle. Now that we know the slope, we're one step closer to finding the equation of the parallel line that passes through the point (4,3).

Using the Point-Slope Form to Find the Equation

Alright, so we've nailed down the slope, which is 1. Now, how do we actually find the equation of the line that passes through the point (4,3) and has this slope? This is where another handy form comes into play: the point-slope form. This form is perfect for situations where you know a point on the line and the slope. It looks like this: y - y1 = m(x - x1), where (x1, y1) is a point on the line and 'm' is the slope. We know our slope (m = 1) and a point on the line (4,3), so we can plug these values directly into the point-slope form. This gives us y - 3 = 1(x - 4). Now, let's simplify this equation. We can distribute the 1 on the right side, which doesn't change anything in this case, so we have y - 3 = x - 4. To get the equation into slope-intercept form (which is often a nice way to present the final answer), we can add 3 to both sides. This gives us y = x - 1. And there you have it! We've found one of the equations of a line that passes through (4,3) and is parallel to 2x - 2y = 11. It's y = x - 1. But hold on a second... the problem asked for the equations of the lines. Plural. Does that mean there's more than one solution? Well, in a way, yes. While there's only one unique line that passes through (4,3) and is parallel to 2x - 2y = 11, we can express this line in different forms. We've already found the slope-intercept form (y = x - 1), but we can also write it in standard form.

Converting to Standard Form and Understanding the Solution

So, we've got our equation in slope-intercept form, y = x - 1. Nice and clean! But sometimes, you might want to express your answer in standard form, which looks like Ax + By = C, where A, B, and C are integers, and A is usually positive. Let's convert our equation to standard form. To do this, we want to get the 'x' and 'y' terms on the same side of the equation and the constant term on the other side. We can subtract 'x' from both sides of y = x - 1, which gives us -x + y = -1. Now, to make 'A' positive, we can multiply the entire equation by -1. This gives us x - y = 1. And there you have it! The equation of the line in standard form is x - y = 1. So, even though we have two different-looking equations (y = x - 1 and x - y = 1), they both represent the same line. They're just different ways of expressing the same mathematical relationship. Now, let's take a step back and think about what we've actually accomplished. We started with a point (4,3) and a line (2x - 2y = 11) and found the equation of a line that's parallel to the given line and passes through the given point. We did this by understanding the relationship between parallel lines and their slopes, using the slope-intercept form to find the slope of the given line, and then using the point-slope form to find the equation of the parallel line. We also learned how to convert between slope-intercept form and standard form. This is a powerful set of tools that you can use to solve all sorts of problems in coordinate geometry. Remember, the key is to break down the problem into smaller, manageable steps and to use the right tools for the job. With practice, you'll become a pro at finding equations of lines in no time! Keep up the great work, guys! And never stop exploring the wonderful world of math!

Find the equations of the line(s) that pass through the point (4, 3) and are parallel to the line 2x - 2y = 11.

Finding Equations of Parallel Lines A Comprehensive Guide