Parallel Line Equation: Step-by-Step Solution

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Hey guys! Today, we're going to tackle a classic math problem: finding the equation of a line that's parallel to another line and passes through a specific point. This might sound intimidating, but trust me, it's totally manageable once we break it down. We'll use the given equation 6y + 3x = -4 and the point (-3, 4) to find our new equation in the standard form ax + by = c. So, grab your pencils and let’s dive in!

Understanding Parallel Lines

Before we jump into the calculations, let's make sure we're all on the same page about what parallel lines actually are. Parallel lines are lines that run in the same direction and never intersect. Think of railroad tracks – they go on forever without ever crossing each other. The key characteristic of parallel lines is that they have the same slope. This is super important because the slope is what determines the direction of a line. If two lines have the same slope, they're going in the same direction, and therefore, they're parallel.

In mathematical terms, the slope of a line is represented by the letter m in the slope-intercept form of a line equation, which is y = mx + b, where b is the y-intercept. The slope tells us how much the line rises (or falls) for every unit we move to the right. So, our first task is to figure out the slope of the given line, 6y + 3x = -4. Once we know that, we'll know the slope of our parallel line too. Knowing the slope and a point, we can then construct the equation of the line. It sounds like a plan, right? So, let's begin by finding the slope of the given line.

To find the slope, we need to rearrange the given equation into the slope-intercept form (y = mx + b). This involves isolating y on one side of the equation. Don't worry, it's just a bit of algebraic manipulation, and we can do it together. This step is the foundation of solving this problem, and once we nail it, the rest will fall into place quite smoothly. Let's move on to the next section where we will get our hands dirty with the algebra and transform our equation!

Finding the Slope of the Given Line

Alright, let's get our hands dirty with some algebra! Our given equation is 6y + 3x = -4. To find the slope, we need to get this equation into the slope-intercept form, which, as we discussed, is y = mx + b. Remember, m is the slope, and that's what we're after. So, our goal is to isolate y on the left side of the equation.

First, we need to get rid of the 3x term. We can do this by subtracting 3x from both sides of the equation. This keeps the equation balanced – whatever we do to one side, we have to do to the other. So, we have:

6y + 3x - 3x = -4 - 3x

This simplifies to:

6y = -3x - 4

Now, we're getting closer! We have 6y on the left side, but we want just y. To do that, we need to divide both sides of the equation by 6. Again, we're keeping things balanced:

(6y) / 6 = (-3x - 4) / 6

This simplifies to:

y = (-3/6)x - (4/6)

We can simplify the fractions further. -3/6 simplifies to -1/2, and -4/6 simplifies to -2/3. So, our equation now looks like this:

y = (-1/2)x - 2/3

Boom! We've done it! We've successfully transformed our equation into slope-intercept form. Now, we can clearly see the slope. The slope, m, is the coefficient of x, which in this case is -1/2. Remember, parallel lines have the same slope. So, the line we're trying to find also has a slope of -1/2. Having the slope is a massive step forward. Next, we'll use this slope and the given point to figure out the full equation of our parallel line. Stay with me, we're making great progress!

Using Point-Slope Form

Okay, now we know the slope of our parallel line is -1/2, and we also know it passes through the point (-3, 4). How do we use this information to find the equation of the line? This is where the point-slope form of a line equation comes to the rescue! The point-slope form is super handy when you have a point and a slope, and it looks like this:

y - y1 = m(x - x1)

Where:

  • m is the slope of the line.
  • (x1, y1) is a point on the line.

We have all this information! We know m = -1/2 and (x1, y1) = (-3, 4). So, let's plug these values into the point-slope form:

y - 4 = (-1/2)(x - (-3))

Notice the double negative in the (x - (-3)) part. Subtracting a negative is the same as adding, so we can rewrite this as:

y - 4 = (-1/2)(x + 3)

Now, we have an equation! But it's not quite in the form we want (ax + by = c). We need to do a little more simplifying and rearranging. The first step is to distribute the -1/2 on the right side of the equation. This means multiplying both x and 3 by -1/2. After we do that, we'll have a cleaner equation to work with, and we'll be one step closer to our final answer. Let's move on and simplify this equation!

Simplifying and Rearranging the Equation

Let's pick up where we left off. We have the equation:

y - 4 = (-1/2)(x + 3)

Our first step is to distribute the -1/2 on the right side. This means multiplying both x and 3 by -1/2:

y - 4 = (-1/2)x + (-1/2)(3)

This simplifies to:

y - 4 = (-1/2)x - 3/2

Now, we want to get the equation into the form ax + by = c. This means we need to get the x and y terms on the same side of the equation and the constant term on the other side. Let's start by adding (1/2)x to both sides:

(1/2)x + y - 4 = (-1/2)x + (1/2)x - 3/2

This simplifies to:

(1/2)x + y - 4 = -3/2

Next, let's get rid of the -4 on the left side by adding 4 to both sides:

(1/2)x + y - 4 + 4 = -3/2 + 4

This simplifies to:

(1/2)x + y = -3/2 + 4

Now we need to combine the constant terms on the right side. To do this, we need a common denominator. We can rewrite 4 as 8/2:

(1/2)x + y = -3/2 + 8/2

This simplifies to:

(1/2)x + y = 5/2

We're almost there! We have the equation in the form ax + by = c, but it looks a little messy with the fractions. To get rid of the fractions, we can multiply both sides of the equation by 2. This will clear out the denominators and give us a cleaner final answer. Let's do that in the next step!

Eliminating Fractions and Final Answer

We're in the home stretch now! Our equation looks like this:

(1/2)x + y = 5/2

To get rid of the fractions, we're going to multiply both sides of the equation by 2. This is a common trick to clean up equations and make them look nicer. So, let's do it:

2 * ((1/2)x + y) = 2 * (5/2)

We need to distribute the 2 on the left side:

2 * (1/2)x + 2 * y = 2 * (5/2)

This simplifies to:

x + 2y = 5

Ta-da! We've done it! We've successfully found the equation of the line parallel to 6y + 3x = -4 and passing through the point (-3, 4). Our final answer, in the form ax + by = c, is:

x + 2y = 5

That wasn't so bad, was it? We broke it down step by step, and now we have a clear and concise equation. Remember, the key was to find the slope of the original line, use that slope for our parallel line, and then use the point-slope form to build our equation. Congratulations on making it through this problem! You've added another tool to your math toolkit. If you follow these steps carefully, you can solve similar problems with ease. Now, you can confidently tackle other parallel line problems. Keep practicing, and you'll become a pro in no time!