Parallel Lines: Find The Equation Passing Through Origin

Hey everyone! Today, we're diving into a fun geometry problem: finding the equation of a line that's parallel to another line and passes through the origin. Let's break it down step-by-step so you can ace these types of questions.

Understanding the Problem

So, the main goal here is understanding how to find the equation of a line. We've got a line, let's call it $AB$, that goes through two points: $A(-3, 0)$ and $B(-6, 5)$. Our mission, should we choose to accept it, is to find the equation of another line that is parallel to $AB$ and, here's the kicker, passes through the origin $(0, 0)$.

Now, why is this important? Well, understanding parallel lines and their equations is a fundamental concept in coordinate geometry. This pops up everywhere, from basic math classes to more advanced physics and engineering problems. Being able to quickly and accurately solve these problems is a seriously valuable skill.

Key Concepts to Keep in Mind

Before we jump into the solution, let’s refresh a couple of key concepts. First off, parallel lines. Remember, parallel lines are lines that run in the same direction and never intersect. The big takeaway here is that parallel lines have the same slope. This is super important. Got it? Good!

Secondly, let’s think about the slope-intercept form of a line equation. You've probably seen this before: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept (the point where the line crosses the y-axis). When a line passes through the origin, that y-intercept, $b$, is zero. So, our equation simplifies to $y = mx$. This makes our job a little easier.

We will first find the slope of line $AB$, we’ll use that same slope for our new line because it's parallel. Then, since we know our new line passes through the origin, we can plug in the slope and the point $(0, 0)$ to nail down the equation.

Step-by-Step Solution

Okay, let's roll up our sleeves and get into the solution. We're gonna break this down into manageable chunks.

1. Calculate the Slope of Line $AB$

The very first thing we need to do is figure out the slope of the line $AB$. Remember the slope formula? It’s the change in $y$ divided by the change in $x$ (rise over run). Mathematically, it’s expressed as:

$m = (y_2 - y_1) / (x_2 - x_1)$

We’ve got our points $A(-3, 0)$ and $B(-6, 5)$. Let’s plug those values into our formula. We’ll call $A$ our $(x_1, y_1)$ and $B$ our $(x_2, y_2)$.

So we get:

$m = (5 - 0) / (-6 - (-3))$

Simplify that, and we have:

$m = 5 / (-6 + 3)$

$m = 5 / -3$

$m = -5/3$

Alright! We've got our slope for line $AB$. It’s $-5/3$. This means that for every 3 units we move to the right on the line, we move 5 units down. Knowing the slope is crucial, as we'll use it to determine the equation of the parallel line. Remember, parallel lines share the same slope, so this value is key to solving our problem.

2. Determine the Slope of the Parallel Line

Now, remember what we said about parallel lines? They have the same slope. This is the golden ticket, guys! Since our new line is parallel to $AB$, it will have the exact same slope. So, the slope of our parallel line is also $-5/3$. See how easy that was?

This concept is super important in coordinate geometry. If you ever need to find the equation of a line parallel to another, just remember their slopes are identical. This shortcut can save you a ton of time and effort, especially on tests!

3. Use the Slope and the Origin to Find the Equation

We're on the home stretch now! We know the slope of our parallel line is $-5/3$, and we know it passes through the origin $(0, 0)$. Remember our simplified slope-intercept form for lines passing through the origin? It’s $y = mx$, where $m$ is the slope.

Let’s plug in our slope, $m = -5/3$, into the equation:

$y = (-5/3)x$

Now, let’s get rid of that fraction to make our equation look a little cleaner. We can do this by multiplying both sides of the equation by 3:

$3y = -5x$

Next, let’s rearrange the equation so that both the $x$ and $y$ terms are on the same side. We’ll add $5x$ to both sides:

$5x + 3y = 0$

And there you have it! We have the equation of our line in standard form. This is a common way to present linear equations, and it makes it easy to compare our answer to the options given.

Matching the Solution to the Answer Choices

Alright, let's peek at those answer choices and see which one matches our equation:

A. $5x - 3y = 0$ B. $-x + 3y = 0$ C. $-5x - 3y = 0$

Our equation is $5x + 3y = 0$. Looking at the options, we see that none of them exactly match. But wait! Notice that option A, $5x - 3y = 0$, is close. It just has a different sign in front of the $3y$ term. Our equation has $+3y$, while option A has $-3y$. So, option A isn't the right answer.

Option C, $-5x - 3y = 0$, has the wrong signs for both terms. It looks like the signs have been flipped compared to our equation, so this one's out too.

But let's think a bit more… if we multiply both sides of our equation $5x + 3y = 0$ by $-1$, we get:

$-5x - 3y = 0$

Whoops! This actually matches option C! It looks like there might have been a slight error in our steps, or perhaps in the original options. It’s always good to double-check our work.

Let's backtrack and review our calculations to make sure we didn’t make a mistake along the way. This is a critical step in problem-solving, as it helps us identify and correct any errors, ensuring we arrive at the correct answer.

Double-Checking for Errors

Okay, let’s go back to our slope calculation. We had points $A(-3, 0)$ and $B(-6, 5)$, and we used the slope formula:

$m = (y_2 - y_1) / (x_2 - x_1)$

Plugging in our values:

$m = (5 - 0) / (-6 - (-3))$

$m = 5 / (-6 + 3)$

$m = 5 / -3$

$m = -5/3$

The slope calculation looks solid. We got $-5/3$, which makes sense.

Then, we used this slope for our parallel line, which is correct because parallel lines have the same slope. We plugged the slope into the equation $y = mx$:

$y = (-5/3)x$

Multiplying both sides by 3:

$3y = -5x$

Adding $5x$ to both sides:

$5x + 3y = 0$

Hmm, our steps still look correct. So, our equation should be $5x + 3y = 0$. However, none of the provided answer choices perfectly match this. Option C, $-5x - 3y = 0$, is equivalent if we multiply our equation by $-1$, but that's not explicitly $5x + 3y = 0$. This suggests there might be an issue with the answer options themselves.

Important Note: In a real test situation, if you're confident in your work and your answer doesn't perfectly match, choose the closest option and make a note to review it later if you have time. Sometimes there are errors in the answer keys, and bringing it to the attention of the instructor can help.

Final Answer and Takeaways

Based on our calculations, the equation of the line that passes through the origin and is parallel to line $AB$ is $5x + 3y = 0$. However, looking at the provided options, the closest one is C. $-5x - 3y = 0$. This highlights the importance of double-checking your work and understanding that sometimes errors can occur in the given choices.

Key Takeaways From This Problem

• Parallel Lines, Same Slope: Parallel lines have the same slope. This is a fundamental concept that’s crucial for solving these types of problems.
• Slope-Intercept Form: The equation $y = mx + b$ is your friend. When a line passes through the origin, it simplifies to $y = mx$.
• Slope Formula: Mastering the slope formula $m = (y_2 - y_1) / (x_2 - x_1)$ is essential.
• Don't Be Afraid to Double-Check: Always, always double-check your work. It can save you from making mistakes, even if the answer choices are tricky.
• Understanding Equivalent Equations: Recognizing that equations can look different but still represent the same line (like $5x + 3y = 0$ and $-5x - 3y = 0$) is a valuable skill.

So, there you have it! We tackled a geometry problem, found the equation of a parallel line, and learned some valuable problem-solving tips along the way. Keep practicing, and you'll be a pro at these in no time!