Hey guys! Let's dive into a super interesting problem involving linear equations and parallel lines. This is a classic topic in algebra, and understanding it can really boost your math skills. We're going to break down a problem step-by-step, making sure everything is crystal clear. So, grab your thinking caps, and let's get started!
Understanding the Basics of Linear Equations
Before we jump into the problem, let's quickly refresh our understanding of linear equations. A linear equation, in its simplest form, represents a straight line on a graph. The standard form of a linear equation is y = mx + b, where:
- m is the slope of the line, indicating its steepness and direction.
- b is the y-intercept, which is the point where the line crosses the y-axis.
The slope, m, is super important because it tells us how much the line rises or falls for every unit we move to the right. A positive slope means the line goes up as we move right, a negative slope means it goes down, a slope of zero means it's a horizontal line, and an undefined slope means it's a vertical line. The y-intercept, b, is the value of y when x is zero. It's the point (0, b) on the graph. Knowing the slope and y-intercept gives us a clear picture of the line's position and direction on the coordinate plane. This understanding is fundamental for solving problems involving lines, especially when we talk about parallel and perpendicular lines.
The Problem: Finding the Equation of a Parallel Line
Okay, here's the problem we're going to tackle:
Line m in the standard (x, y) coordinate plane has the equation 3x - 2y = 8. Line n is parallel to line m and has a y-intercept that is 4 more than the y-intercept of m. Line n has which of the following equations?
This might seem a bit complex at first, but don't worry! We're going to break it down into manageable steps. The key here is understanding the concept of parallel lines and how their equations relate to each other. Parallel lines have the same slope but different y-intercepts. This is crucial because if they had the same slope and y-intercept, they would be the same line, not parallel lines. So, when we look at the equation of line m, we need to figure out its slope and y-intercept, and then use that information to find the equation of line n.
Step 1: Finding the Slope and y-intercept of Line m
So, the first thing we need to do is figure out the slope and y-intercept of line m. The equation for line m is given as 3x - 2y = 8. To make things easier, we want to rewrite this equation in the slope-intercept form (y = mx + b). This will allow us to easily identify the slope and y-intercept.
Let's rearrange the equation:
- Subtract 3x from both sides: -2y = -3x + 8
- Divide both sides by -2: y = (3/2)x - 4
Now, the equation is in the slope-intercept form. We can see that the slope of line m is 3/2, and the y-intercept is -4. Remember, the slope is the coefficient of x, and the y-intercept is the constant term. Knowing these two values is a big step forward. We now know the steepness and the point where line m crosses the y-axis. This information is crucial for finding the equation of line n, which is parallel to m.
Step 2: Determining the y-intercept of Line n
The problem tells us that line n has a y-intercept that is 4 more than the y-intercept of m. We already found that the y-intercept of m is -4. So, to find the y-intercept of n, we simply add 4 to -4:
y-intercept of n = -4 + 4 = 0
So, the y-intercept of line n is 0. This means that line n passes through the origin (0, 0). This is an important piece of information because it helps us narrow down the possible equations for line n. We know that the equation of line n will have the form y = mx + 0, which simplifies to y = mx. Now, we just need to find the slope of line n.
Step 3: Finding the Slope of Line n
Here's where the concept of parallel lines comes into play. Remember, parallel lines have the same slope. We already determined that the slope of line m is 3/2. Since line n is parallel to line m, line n must also have a slope of 3/2. This is a key property of parallel lines, and it makes our job much easier.
Now we know the slope of line n is 3/2, and we also know its y-intercept is 0. We have all the information we need to write the equation of line n. We can simply plug these values into the slope-intercept form (y = mx + b).
Step 4: Writing the Equation of Line n
We know that the slope (m) of line n is 3/2 and the y-intercept (b) is 0. Plugging these values into the slope-intercept form (y = mx + b), we get:
y = (3/2)x + 0
This simplifies to:
y = (3/2)x
So, the equation of line n is y = (3/2)x. This is the final answer, and it matches one of the options provided in the problem. We've successfully found the equation of the line parallel to line m with a y-intercept 4 more than that of line m.
Conclusion: Mastering Linear Equations
Awesome job, guys! We've successfully navigated through this problem and found the equation of line n. We broke it down step-by-step, making sure we understood each concept along the way. Remember, the key to solving problems like this is to:
- Understand the basics of linear equations and the slope-intercept form.
- Identify the slope and y-intercept of the given line.
- Use the properties of parallel lines (same slope) to find the slope of the parallel line.
- Determine the y-intercept of the parallel line based on the given information.
- Write the equation of the parallel line using the slope-intercept form.
By following these steps, you can confidently tackle similar problems involving parallel lines and linear equations. Keep practicing, and you'll become a pro in no time! If you have any questions or want to explore more examples, feel free to ask. Keep up the great work!