Find The Equation Of The Line Which Passes Through The Point \[$(-6,14)\$\] And Is Perpendicular To The Given Line In Slope-intercept Form. Simplify Your Answer.Given Line: \[$4x + 8y = 7y - 3\$\]
Introduction
In mathematics, finding the equation of a line that passes through a given point and is perpendicular to another line is a fundamental concept in geometry and algebra. This problem involves finding the equation of a line that is perpendicular to the given line in slope-intercept form and passes through the point (-6, 14). In this article, we will discuss the steps to find the equation of the perpendicular line and provide a simplified answer.
Understanding the Given Line
The given line is in the form of a linear equation, which is:
4x + 8y = 7y - 3
To find the slope-intercept form of the line, we need to isolate the variable y. We can do this by subtracting 8y from both sides of the equation and then dividing both sides by -8.
4x + 8y - 8y = 7y - 3 - 8y
4x = -y - 3
y = -4x - 3
Now that we have the slope-intercept form of the given line, we can identify the slope (m) and the y-intercept (b). The slope is -4, and the y-intercept is -3.
Finding the Slope of the Perpendicular Line
The slope of the perpendicular line is the negative reciprocal of the slope of the given line. To find the negative reciprocal of -4, we can multiply -4 by -1.
m = -1 / (-4)
m = 1/4
Now that we have the slope of the perpendicular line, we can use it to find the equation of the line.
Finding the Equation of the Perpendicular Line
To find the equation of the perpendicular line, we can use the point-slope form of a linear equation, which is:
y - y1 = m(x - x1)
where (x1, y1) is the given point (-6, 14), and m is the slope of the perpendicular line (1/4).
y - 14 = (1/4)(x - (-6))
y - 14 = (1/4)(x + 6)
y - 14 = (1/4)x + 1.5
y = (1/4)x + 1.5 + 14
y = (1/4)x + 15.5
Now that we have the equation of the perpendicular line, we can simplify it by multiplying both sides by 4 to eliminate the fraction.
4y = x + 62
Conclusion
In this article, we discussed how to find the equation of a line that passes through a given point and is perpendicular to another line. We started by finding the slope-intercept form of the given line and then identified the slope and y-intercept. We then found the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line. Finally, we used the point-slope form of a linear equation to find the equation of the perpendicular line and simplified it by multiplying both sides by 4.
Key Takeaways
- To find the equation of a line that passes through a given point and is perpendicular to another line, we need to find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
- We can use the point-slope form of a linear equation to find the equation of the perpendicular line.
- To simplify the equation of the perpendicular line, we can multiply both sides by a common factor to eliminate any fractions.
Final Answer
The equation of the line that passes through the point (-6, 14) and is perpendicular to the given line is:
4y = x + 62
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the slope of a line given in standard form?
A: To find the slope of a line given in standard form, you need to rewrite the equation in slope-intercept form. To do this, you can use the following steps:
- Subtract the constant term from both sides of the equation.
- Divide both sides of the equation by the coefficient of the x-term.
For example, if the equation is 4x + 8y = 7y - 3, you can rewrite it as:
y = (-4/8)x + (-3/8) y = (-1/2)x - 3/8
Now that the equation is in slope-intercept form, you can identify the slope (m) and the y-intercept (b).
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is the reciprocal of the slope multiplied by -1. For example, if the slope is 2, the negative reciprocal is -1/2.
Q: How do I find the equation of a line that passes through a given point and is perpendicular to another line?
A: To find the equation of a line that passes through a given point and is perpendicular to another line, you need to follow these steps:
- Find the slope of the given line.
- Find the negative reciprocal of the slope of the given line.
- Use the point-slope form of a linear equation to find the equation of the perpendicular line.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Q: How do I simplify the equation of a line?
A: To simplify the equation of a line, you can multiply both sides of the equation by a common factor to eliminate any fractions.
Q: What is the final answer to the problem?
A: The final answer to the problem is the equation of the line that passes through the point (-6, 14) and is perpendicular to the given line, which is:
4y = x + 62
Common Mistakes to Avoid
- Not rewriting the equation in slope-intercept form before finding the slope.
- Not finding the negative reciprocal of the slope of the given line.
- Not using the point-slope form of a linear equation to find the equation of the perpendicular line.
- Not simplifying the equation of the line by multiplying both sides by a common factor.
Conclusion
In this article, we discussed frequently asked questions related to finding the equation of a line that passes through a given point and is perpendicular to another line. We covered topics such as the slope-intercept form of a linear equation, finding the slope of a line given in standard form, the negative reciprocal of a slope, and simplifying the equation of a line. We also provided common mistakes to avoid and a final answer to the problem.