Equation Of A Parallel Line How To Find It
Hey guys! Let's dive into a common math problem: finding the equation of a line that's parallel to another line and passes through a specific point. This is a fundamental concept in coordinate geometry, and mastering it will definitely help you ace your math exams. We'll break down the steps, making it super easy to understand.
Understanding Parallel Lines and Their Slopes
First off, parallel lines are lines that run in the same direction and never intersect. The key characteristic of parallel lines is that they have the same slope. Remember the slope-intercept form of a line: y = mx + b, where m represents the slope and b represents the y-intercept. When we're given a line and need to find a parallel line, the first thing we focus on is the slope. Identifying the slope is the crucial first step in solving these types of problems.
In our problem, the given line is y - 1 = 4(x + 3). To find its slope, we need to rewrite this equation in slope-intercept form (y = mx + b). Let's do that:
y - 1 = 4(x + 3)
Distribute the 4 on the right side:
y - 1 = 4x + 12
Add 1 to both sides to isolate y:
y = 4x + 13
Now we can clearly see that the slope (m) of the given line is 4. Since parallel lines have the same slope, any line parallel to this one will also have a slope of 4. This is a super important point to remember!
Using the Point-Slope Form
Now that we know the slope of our parallel line is 4, we need to find its equation. We're also given that this line passes through the point (4, 32). To find the equation of a line when we know a point and the slope, we use the point-slope form. The point-slope form is a powerful tool, guys, so make sure you're comfortable with it. It's given by:
y - y₁ = m(x - x₁)
Where:
- (x₁, y₁) is the given point
- m is the slope
In our case, (x₁, y₁) = (4, 32) and m = 4. Let's plug these values into the point-slope form:
y - 32 = 4(x - 4)
This is the equation of our line in point-slope form. However, the answer choices are in slope-intercept form (y = mx + b), so we need to convert it. Let's do that in the next step!
Converting to Slope-Intercept Form
To convert our equation from point-slope form to slope-intercept form, we need to distribute the 4 on the right side and then isolate y. Let's take our equation:
y - 32 = 4(x - 4)
First, distribute the 4:
y - 32 = 4x - 16
Now, add 32 to both sides to isolate y:
y = 4x - 16 + 32
Simplify:
y = 4x + 16
Voila! We have our equation in slope-intercept form. The equation of the line parallel to y - 1 = 4(x + 3) and passing through the point (4, 32) is y = 4x + 16. Looking at the answer choices, we see that this corresponds to option D. Awesome, right?
Checking Our Answer
It's always a good idea to double-check your answer, especially in math problems. We can do this in a couple of ways. First, we know that our line has a slope of 4, which is the same as the slope of the given line, so that checks out. Second, we can plug the point (4, 32) into our equation to make sure it satisfies the equation:
y = 4x + 16
32 = 4(4) + 16
32 = 16 + 16
32 = 32
The equation holds true! This confirms that our answer is correct.
Common Mistakes to Avoid
When solving these types of problems, there are a few common mistakes that students often make. Here are some things to watch out for:
- Not correctly identifying the slope of the given line: Make sure you rewrite the equation in slope-intercept form (y = mx + b) to easily see the slope.
- Using the negative reciprocal of the slope: Remember, parallel lines have the same slope, not the negative reciprocal (which is for perpendicular lines).
- Incorrectly applying the point-slope form: Double-check that you're plugging the values into the correct places in the formula.
- Making arithmetic errors: Be careful with your calculations, especially when distributing and simplifying equations.
By avoiding these mistakes, you'll be well on your way to mastering these problems!
Practice Problems
To really solidify your understanding, try solving some more problems like this. Here are a couple you can try:
- Find the equation of the line parallel to y = -2x + 5 and passing through the point (1, -3).
- What is the equation of the line that is parallel to 2y - 6x = 8 and passes through the point (-2, 4)?
Work through these problems step-by-step, and you'll become a pro at finding equations of parallel lines in no time!
Conclusion
Finding the equation of a line parallel to another line is a crucial skill in algebra. By understanding the concept of slope, mastering the point-slope form, and avoiding common mistakes, you can confidently tackle these problems. Remember, the key is to identify the slope of the given line, use that same slope for the parallel line, and then use the point-slope form to find the equation. Keep practicing, and you'll nail it! Remember guys, math is all about practice, so keep at it, and you'll be amazed at what you can achieve. Good luck, and happy solving!
Find the equation of the line parallel to the line y - 1 = 4(x + 3) and passing through the point (4, 32).
Equation of a Parallel Line How to Find It