Line Equation: Parallel Line Through (4,-1)
Finding the equation of a line can seem daunting, but it's totally manageable once you break it down. In this article, we're going to walk through how to find the equation of a line that passes through a specific point and is parallel to another given line. Specifically, we're tackling the problem: Find the equation of a line that passes through the point (4, -1) and is parallel to the line y = (1/2)x + 3. So, let's dive in and make it super clear!
Understanding Parallel Lines
Before we jump into the math, let's quickly recap what it means for lines to be parallel. Parallel lines are lines that run in the same direction and never intersect. The most important thing to remember about parallel lines is that they have the same slope. This is the key to solving our problem.
In our case, the given line is y = (1/2)x + 3. This equation is in slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. So, for the line y = (1/2)x + 3, the slope (m) is 1/2. Since we want a line parallel to this one, our new line will also have a slope of 1/2. Got it? Great, let's move on!
Knowing the slope is the first crucial step. When lines are parallel, they share the same slope, meaning they have the same steepness and direction. Think of train tracks; they run side by side, never meeting, always maintaining the same angle relative to the horizontal. Mathematically, this characteristic simplifies things considerably. It allows us to take the slope from one line and directly apply it to another line that we want to be parallel. It's this principle that allows us to confidently say that our new line, which is parallel to y = (1/2)x + 3, also has a slope of 1/2. This concept is vital not only in coordinate geometry but also in various applications in physics and engineering, where understanding the relationships between parallel paths or forces is important. Furthermore, recognizing parallel lines can significantly aid in solving more complex geometric problems involving shapes and spatial reasoning. So, remember: parallel lines mean equal slopes!
Point-Slope Form
Now that we know the slope of our new line (which is 1/2), and we have a point it passes through (4, -1), we can use the point-slope form of a linear equation. The point-slope form is: y - y1 = m(x - x1), where:
- (x1, y1) is the given point
- m is the slope
In our problem:
- x1 = 4
- y1 = -1
- m = 1/2
Plug these values into the point-slope form:
y - (-1) = (1/2)(x - 4)
Simplify:
y + 1 = (1/2)(x - 4)
Converting to Slope-Intercept Form
While the point-slope form is perfectly valid, it's often useful to convert the equation to slope-intercept form (y = mx + b) to make it easier to compare with other lines and to quickly identify the y-intercept.
Let's take our equation from the point-slope form: y + 1 = (1/2)(x - 4).
Distribute the 1/2 on the right side:
y + 1 = (1/2)x - 2
Subtract 1 from both sides to solve for y:
y = (1/2)x - 2 - 1
Simplify:
y = (1/2)x - 3
So, the equation of the line that passes through the point (4, -1) and is parallel to the line y = (1/2)x + 3 is y = (1/2)x - 3.
Converting to slope-intercept form not only makes it easier to read off the slope and y-intercept, but it also prepares the equation for further analysis or comparison. For instance, if you wanted to graph the line, knowing the y-intercept (-3 in this case) provides a straightforward starting point on the y-axis. Additionally, the slope-intercept form is handy for quickly determining how the line will behave as x changes – for every increase of 1 in x, y increases by 1/2. This kind of understanding is crucial in many real-world applications, such as in economics where you might be analyzing cost functions or in physics where you're looking at linear motion. The algebraic manipulation involved in converting from point-slope to slope-intercept form also reinforces essential mathematical skills that are applicable in more complex problem-solving scenarios. So mastering this conversion is a valuable tool in your mathematical toolkit.
Verification
To make sure we did everything correctly, let's quickly verify that our new line, y = (1/2)x - 3, indeed passes through the point (4, -1).
Plug in x = 4 into our equation:
y = (1/2)(4) - 3
y = 2 - 3
y = -1
Since we got y = -1 when x = 4, our line does indeed pass through the point (4, -1). Also, we already know it's parallel because it has the same slope as the given line.
Verifying our solution is a critical step in problem-solving. By substituting the given point into our derived equation, we confirm that the equation holds true for that specific coordinate. This process not only ensures that we haven't made any algebraic errors along the way but also reinforces our understanding of what an equation of a line represents – a relationship between x and y coordinates that is satisfied by all points on the line. In this case, plugging in x = 4 should yield y = -1 if our equation is correct. Furthermore, verification steps can reveal if we've misinterpreted the problem or overlooked some crucial detail. It acts as a safety net, catching potential mistakes before they lead to incorrect conclusions. This practice is beneficial not only in mathematics but in many fields where accuracy is essential. Always double-check your work; it pays off!
Alternative Method: Slope-Intercept Form Directly
Another way to approach this problem is to directly use the slope-intercept form (y = mx + b) from the start.
We know the slope (m) is 1/2, so our equation will look like this:
y = (1/2)x + b
Now, we need to find the y-intercept (b). We know the line passes through the point (4, -1), so we can plug in these values for x and y:
-1 = (1/2)(4) + b
Simplify:
-1 = 2 + b
Solve for b:
b = -1 - 2
b = -3
So, our equation is:
y = (1/2)x - 3
This method arrives at the same answer as the point-slope method, just through a slightly different approach.
Using the slope-intercept form directly can sometimes be more intuitive for those who are very comfortable with that form. By substituting the known slope and the coordinates of the point into y = mx + b, we effectively solve for the y-intercept. This approach can be particularly useful when the y-intercept is the primary unknown we need to find. However, it's worth noting that this method requires a bit more upfront calculation compared to the point-slope form, especially if the given point has non-zero coordinates. Both methods are equally valid and lead to the correct answer, but choosing one over the other might depend on personal preference or the specific details of the problem at hand. The key is to understand both approaches and feel comfortable using whichever one suits you best.
Conclusion
Finding the equation of a line that satisfies certain conditions might seem tricky at first, but with a solid understanding of the underlying principles—like what parallel lines are and how to use the point-slope and slope-intercept forms—it becomes much more straightforward. We successfully found that the equation of the line passing through (4, -1) and parallel to y = (1/2)x + 3 is y = (1/2)x - 3. Keep practicing, and you'll master these concepts in no time! Remember practice makes perfect, so keep at it!