Parallel Line Equation With X-intercept Of -3
Hey guys! Let's dive into finding the equation of a line that's parallel to another line, but with a specific twist – it has an x-intercept of -3. This is a classic problem in algebra and understanding it can really boost your skills in coordinate geometry. We'll break it down step-by-step, making sure it's super clear and easy to follow. So, let’s jump right into it!
Understanding Parallel Lines and Their Equations
When we talk about parallel lines, the most important thing to remember is that they have the same slope. That’s right, identical slopes. Think of it like this: parallel lines are like train tracks – they run alongside each other without ever meeting. Mathematically, this means their steepness (slope) is the same. The slope of a line tells us how much the line rises (or falls) for every unit it runs horizontally. If two lines have the same 'rise over run,' they'll never intersect.
The equation of a line is typically expressed in slope-intercept form, which is y = mx + b. Here, 'm' represents the slope, and 'b' represents the y-intercept (the point where the line crosses the y-axis). Knowing this form is crucial because it gives us a direct way to read off the slope and y-intercept of any line, just by looking at its equation. For our problem, identifying the slope is the first key step because any line parallel to the given line will share the same slope value. So, whenever you see a problem mentioning parallel lines, your mind should immediately jump to: “Same slope!”
Now, let's consider the x-intercept. The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. This is a super helpful piece of information because it gives us a specific point (x, 0) that our line passes through. In our case, we know the x-intercept is -3, which means the line passes through the point (-3, 0). We can use this point, along with the slope we determined from the parallel line condition, to find the full equation of our line. Combining the concept of parallel slopes and the x-intercept gives us a powerful way to nail this kind of problem. We'll use these concepts in the next sections to solve our specific problem and show you how it’s done!
Identifying the Slope from the Given Options
Okay, let's get practical and look at the options provided to us. We need to figure out which equation represents a line parallel to a given line and has an x-intercept of -3. Remember, the first thing we need to find is the slope. The slope is crucial because parallel lines share the same slope. To identify the slope, we need to look at the y = mx + b form, where 'm' is the slope. Let’s examine the options:
A. y = (2/3)x + 3
B. y = (2/3)x + 2
C. y = (-3/2)x + 3
D. y = (-3/2)x + 2
In options A and B, the slope (the coefficient of x) is 2/3. This means these two lines have the same steepness and are parallel to each other. However, we don’t know yet if either of these is the line we're looking for. We still need to check the x-intercept condition. Options C and D have a slope of -3/2. These lines are also parallel to each other, but they have a different steepness than the lines in options A and B. So, right off the bat, we know that the line we are looking for will either have a slope of 2/3 or -3/2. The trick now is to figure out which slope, along with the x-intercept condition, gives us the correct equation.
To do this, we can use the point-slope form of a line equation, which is y - y1 = m(x - x1). This form is super useful when we know the slope (m) and a point on the line (x1, y1). In our case, we know the x-intercept is -3, so the point is (-3, 0). We'll plug this point into the equation along with each possible slope to see which one works. This is where the fun begins, as we get to use our algebra skills to narrow down the correct answer. By systematically checking each option, we can confidently identify the line that meets both the parallel slope condition and the x-intercept condition. So, let's move on and test each slope with the given x-intercept!
Verifying the X-intercept Condition
Alright, let’s put our detective hats on and verify which of the options actually has an x-intercept of -3. Remember, the x-intercept is the point where the line crosses the x-axis, and at this point, y is always 0. So, to check if a line has an x-intercept of -3, we plug in x = -3 and y = 0 into the equation and see if it holds true. If it does, bingo! That's our line.
Let's start with option A: y = (2/3)x + 3. Plug in x = -3 and y = 0:
0 = (2/3)(-3) + 3
0 = -2 + 3
0 = 1 This is not true. So, option A doesn't have an x-intercept of -3. Bummer, but on to the next!
Now let's try option B: y = (2/3)x + 2. Plug in x = -3 and y = 0:
0 = (2/3)(-3) + 2
0 = -2 + 2
0 = 0 This is true! Option B checks out. Before we get too excited, let's make sure the other options don't work, just to be absolutely sure.
Let's check option C: y = (-3/2)x + 3. Plug in x = -3 and y = 0:
0 = (-3/2)(-3) + 3
0 = (9/2) + 3
0 = 4.5 + 3 This is definitely not true. Option C is out.
Finally, let’s look at option D: y = (-3/2)x + 2. Plug in x = -3 and y = 0:
0 = (-3/2)(-3) + 2
0 = (9/2) + 2
0 = 4.5 + 2 Nope, not true either. Option D is also out.
So, after carefully checking all the options, we found that only option B satisfies the condition of having an x-intercept of -3. This means we're one step closer to our final answer! The process of verifying each option might seem a bit tedious, but it’s a surefire way to make sure we get the correct equation. Plus, it reinforces our understanding of x-intercepts and how they relate to the equation of a line. Now that we've identified the correct x-intercept, let's put it all together and confidently state our solution.
The Final Answer
Alright, guys, after carefully analyzing the slopes and verifying the x-intercept condition, we’ve arrived at the final answer! Remember, we were looking for the equation of a line that is parallel to a given line and has an x-intercept of -3. We went through each option, checking the slopes and then plugging in the x-intercept to see which equation holds true.
We found that option B, y = (2/3)x + 2, is the winner! It has a slope that could potentially be parallel to another line (we'd need the original line's equation to confirm the parallel part), and most importantly, it satisfies the x-intercept condition. When we plugged in x = -3 and y = 0, the equation held true, confirming that this line indeed crosses the x-axis at -3.
Therefore, the equation of the line that is parallel to the given line (assuming the given line has a slope of 2/3) and has an x-intercept of -3 is:
B. y = (2/3)x + 2
And there you have it! We've successfully solved the problem by breaking it down into manageable steps. We first understood the concept of parallel lines and their slopes, then we identified the importance of the x-intercept. By systematically checking each option, we were able to confidently arrive at the correct answer. This problem showcases how understanding the fundamental concepts of coordinate geometry can help you tackle even seemingly complex questions. So keep practicing, keep exploring, and you'll become a pro at solving these kinds of problems in no time!