What Is The Slope-intercept Form Of The Equation Of The Line That Passes Through The Point $(-6, 1)$ And Is Perpendicular To The Graph Of $2x + 3y = -5$?A. $y = -\frac{3}{2}x - 8$ B. $y = -\frac{3}{2}x +
Introduction to Slope-Intercept Form
The slope-intercept form of a linear equation is a fundamental concept in mathematics, particularly in algebra and geometry. It is a way to express a linear equation in the form of y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope-intercept form is essential in graphing lines, finding the equation of a line that passes through a given point, and determining the relationship between two lines.
Understanding the Problem
In this problem, we are given a point (-6, 1) and a linear equation 2x + 3y = -5. We need to find the slope-intercept form of the equation of the line that passes through the given point and is perpendicular to the graph of the given linear equation. To solve this problem, we will first find the slope of the given linear equation, then find the slope of the perpendicular line, and finally use the point-slope form to find the equation of the perpendicular line.
Finding the Slope of the Given Linear Equation
The given linear equation is 2x + 3y = -5. To find the slope, we need to rewrite the equation in the slope-intercept form. We can do this by isolating y on one side of the equation.
2x + 3y = -5
3y = -2x - 5
y = -\frac{2}{3}x - \frac{5}{3}
From the rewritten equation, we can see that the slope of the given linear equation is -\frac{2}{3}.
Finding the Slope of the Perpendicular Line
Since the line we are looking for is perpendicular to the given linear equation, its slope will be the negative reciprocal of the slope of the given linear equation. The negative reciprocal of -\frac{2}{3} is \frac{3}{2}.
Using the Point-Slope Form to Find the Equation of the Perpendicular Line
Now that we have the slope of the perpendicular line, we can use the point-slope form to find its equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
We are given the point (-6, 1), so we can substitute x1 = -6 and y1 = 1 into the point-slope form.
y - 1 = \frac{3}{2}(x - (-6))
y - 1 = \frac{3}{2}(x + 6)
y - 1 = \frac{3}{2}x + 9
y = \frac{3}{2}x + 10
Conclusion
In this problem, we found the slope-intercept form of the equation of the line that passes through the point (-6, 1) and is perpendicular to the graph of 2x + 3y = -5. The equation of the perpendicular line is y = \frac{3}{2}x + 10.
Discussion
The slope-intercept form is a powerful tool in mathematics, particularly in algebra and geometry. It allows us to express a linear equation in a simple and intuitive way, making it easier to graph lines, find the equation of a line that passes through a given point, and determine the relationship between two lines.
In this problem, we used the slope-intercept form to find the equation of a line that is perpendicular to another line. This is a common problem in mathematics, and the slope-intercept form is an essential tool in solving it.
Final Answer
The final answer is y = \frac{3}{2}x + 10.
Introduction
The slope-intercept form of a linear equation is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will answer some frequently asked questions about slope-intercept form, including how to find the slope and y-intercept, how to graph lines, and how to determine the relationship between two lines.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is a way to express a linear equation in the form of y = mx + b, where m represents the slope of the line and b represents the y-intercept.
Q: How do I find the slope of a linear equation?
A: To find the slope of a linear equation, you need to rewrite the equation in the slope-intercept form. You can do this by isolating y on one side of the equation.
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the point where the line intersects the y-axis. It is represented by the value of b in the slope-intercept form.
Q: How do I graph a line using the slope-intercept form?
A: To graph a line using the slope-intercept form, you need to find the y-intercept and the slope. You can then use a graphing tool or draw a line on a coordinate plane to represent the equation.
Q: What is the relationship between two lines with the same slope?
A: Two lines with the same slope are parallel to each other. They will never intersect, no matter how far you extend the lines.
Q: What is the relationship between two lines with different slopes?
A: Two lines with different slopes will intersect at a single point. The point of intersection will be the solution to the system of equations.
Q: How do I find the equation of a line that is perpendicular to another line?
A: To find the equation of a line that is perpendicular to another line, you need to find the slope of the perpendicular line. The slope of the perpendicular line will be the negative reciprocal of the slope of the original line.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is the slope that is opposite in sign and reciprocal in value. For example, the negative reciprocal of 2 is -\frac{1}{2}.
Q: How do I use the point-slope form to find the equation of a line?
A: To use the point-slope form to find the equation of a line, you need to substitute the values of x and y into the equation. You can then simplify the equation to find the slope-intercept form.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is a way to express a linear equation in the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
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 find the slope of the perpendicular line and use the point-slope form to find the equation.
Q: What is the final answer to the problem of finding the equation of a line that passes through a given point and is perpendicular to another line?
A: The final answer is y = \frac{3}{2}x + 10.
Conclusion
In this article, we have answered some frequently asked questions about slope-intercept form, including how to find the slope and y-intercept, how to graph lines, and how to determine the relationship between two lines. We have also provided examples and explanations to help you understand the concepts.