Find Equations Of Parallel Line Passing Through (-4, -2)
Hey guys! Let's dive into a cool math problem today. We're going to figure out which equations represent a line that's parallel to the line given by the equation $3x - 4y = 7$ and also passes through the point $(-4, -2)$. Sounds like a fun challenge, right? To nail this, we'll need to brush up on our knowledge of parallel lines and how to find the equation of a line. So, let’s get started!
Understanding Parallel Lines
First things first, what does it mean for two lines to be parallel? Parallel lines, in simple terms, are lines that run in the same direction and never intersect. Think of train tracks – they run side by side and maintain the same distance from each other. The key characteristic of parallel lines that we need to remember is that they have the same slope. The slope of a line tells us how steep it is – basically, how much it rises or falls for every unit it moves horizontally. So, if we want to find a line parallel to $3x - 4y = 7$, the first thing we need to do is figure out the slope of this line. To do that, we'll rewrite the equation in slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Let’s rewrite our equation step by step. We have $3x - 4y = 7$. Our goal is to isolate $y$ on one side of the equation. First, we'll subtract $3x$ from both sides, which gives us $-4y = -3x + 7$. Next, we'll divide both sides by $-4$ to solve for $y$. This gives us $y = \frac{-3}{-4}x + \frac{7}{-4}$, which simplifies to $y = \frac{3}{4}x - \frac{7}{4}$. Now, we can clearly see that the slope of the line $3x - 4y = 7$ is $\frac{3}{4}$. Remember, any line parallel to this one will also have a slope of $\frac{3}{4}$. This is a crucial piece of information that will help us narrow down our options. Now that we know the slope, we need to find the equation of a line with this slope that passes through the point $(-4, -2)$. Let's move on to the next step and figure out how to do that!
Finding the Equation of the Parallel Line
Now that we know the slope of our parallel line needs to be $\frac{3}{4}$, we're halfway there! We also know that this line has to pass through the point $(-4, -2)$. To find the equation of this line, we can use the point-slope form of a linear equation. The point-slope form is a super handy tool that allows us to write the equation of a line if we know its slope and a point it passes through. The formula for the point-slope form is $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the given point. In our case, we have $m = \frac{3}{4}$ and the point $(-4, -2)$. So, we can plug these values into the point-slope form. Substituting the values, we get $y - (-2) = \frac{3}{4}(x - (-4))$. Let's simplify this step by step. First, we can rewrite $y - (-2)$ as $y + 2$ and $x - (-4)$ as $x + 4$. So, our equation becomes $y + 2 = \frac{3}{4}(x + 4)$. Now, we need to distribute the $\frac{3}{4}$ on the right side of the equation. This gives us $y + 2 = \frac{3}{4}x + \frac{3}{4} \cdot 4$, which simplifies to $y + 2 = \frac{3}{4}x + 3$. To get the equation in slope-intercept form ($y = mx + b$), we need to isolate $y$. We can do this by subtracting $2$ from both sides of the equation. This gives us $y = \frac{3}{4}x + 3 - 2$, which simplifies to $y = \frac{3}{4}x + 1$. So, one equation of the line parallel to $3x - 4y = 7$ and passing through $(-4, -2)$ is $y = \frac{3}{4}x + 1$. But wait, the question asks us to select two options. This means we need to find another equation that represents the same line. Remember, there are many ways to write the equation of a line, even if they all represent the same line graphically. One common way is to rewrite the equation in standard form, which is $Ax + By = C$, where $A$, $B$, and $C$ are constants. Let's convert our equation $y = \frac{3}{4}x + 1$ into standard form. First, we want to get rid of the fraction. To do this, we can multiply both sides of the equation by $4$. This gives us $4y = 4(\frac{3}{4}x + 1)$, which simplifies to $4y = 3x + 4$. Now, we want to get the $x$ and $y$ terms on the same side of the equation. We can do this by subtracting $3x$ from both sides. This gives us $-3x + 4y = 4$. To make the coefficient of $x$ positive, we can multiply the entire equation by $-1$. This gives us $3x - 4y = -4$. So, another equation of the line parallel to $3x - 4y = 7$ and passing through $(-4, -2)$ is $3x - 4y = -4$. Now we have two equations that represent the same line, and we’re ready to tackle the given options and select the correct ones.
Selecting the Correct Options
Alright, we've done the hard work! We know that the equations we're looking for must represent a line with a slope of $\frac3}{4}$ and pass through the point $(-4, -2)$. We've even found two equations that fit the bill{4}x + 1$ and $3x - 4y = -4$. Now, let's take a look at the options provided and see which ones match our findings. The options are:
Let's analyze each option one by one.
Option 1: $y = -\frac{3}{4}x + 1$
The slope of this line is $-\frac{3}{4}$, which is not equal to our desired slope of $\frac{3}{4}$. Therefore, this line is not parallel to $3x - 4y = 7$, and we can eliminate this option. It's super important to double-check the slopes, guys, because a simple sign difference can throw the whole thing off!
Option 2: $3x - 4y = -4$
We already derived this equation! We know that this line has a slope of $\frac{3}{4}$ and passes through the point $(-4, -2)$. So, this is definitely one of the correct options. Awesome! We’re on the right track.
Option 3: $4x - 3y = -10$
To determine the slope of this line, we need to rewrite it in slope-intercept form ($y = mx + b$). Let's do that:
First, subtract $4x$ from both sides: $-3y = -4x - 10$
Next, divide both sides by $-3$: $y = \frac{-4}{-3}x + \frac{-10}{-3}$
Simplify: $y = \frac{4}{3}x + \frac{10}{3}$
The slope of this line is $\frac{4}{3}$, which is not equal to our desired slope of $\frac{3}{4}$. So, this line is not parallel to $3x - 4y = 7$, and we can eliminate this option. Phew, that was close!
So, after analyzing all the options, we can confidently say that the only equation that represents a line parallel to $3x - 4y = 7$ and passes through the point $(-4, -2)$ is $3x - 4y = -4$. But remember, we need to select two options. So, what gives? Well, one of the options must be an equivalent form of the equation we found. And we already found that $y = \frac{3}{4}x + 1$ is also a correct equation.
Therefore, the two options that correctly represent the line are $3x - 4y = -4$ and the equation we derived earlier, which is $y = \frac{3}{4}x + 1$.
Conclusion
Great job, everyone! We successfully navigated through this problem by understanding the concept of parallel lines, using the point-slope form, and converting between different forms of linear equations. We found that the equations $3x - 4y = -4$ and $y = \frac{3}{4}x + 1$ represent the line parallel to $3x - 4y = 7$ and passing through the point $(-4, -2)$. This kind of problem really highlights how important it is to have a solid grasp of the fundamentals of linear equations. Keep practicing, and you'll become math wizards in no time! Remember, the key is to break down the problem into smaller, manageable steps and to double-check your work along the way. You guys rock!
Keywords: parallel lines, slope, point-slope form, linear equations, equation of a line, slope-intercept form, standard form, math problem, solving equations, mathematics
