Which Point-slope Equation Represents A Line That Passes Through { (3,-2)$}$ With A Slope Of { -\frac{4}{5}$}$?A. { Y - 3 = -\frac{4}{5}(x + 2)$}$ B. { Y - 2 = \frac{4}{5}(x - 3)$}$ C. [$y + 2 =

Which Point-Slope Equation Represents a Line That Passes Through a Given Point with a Specified Slope?

In mathematics, the point-slope equation is a fundamental concept used to represent a line that passes through a given point with a specified slope. The point-slope equation is a linear equation that takes the form of ${y - y_1 = m(x - x_1)}$, where ${m}$ is the slope of the line, and ${(x_1, y_1)}$ is the given point through which the line passes. In this article, we will explore which point-slope equation represents a line that passes through the point ${(3, -2)}$ with a slope of ${-\frac{4}{5}}$.

Understanding the Point-Slope Equation

The point-slope equation is a linear equation that takes the form of ${y - y_1 = m(x - x_1)}$. Here, ${m}$ is the slope of the line, and ${(x_1, y_1)}$ is the given point through which the line passes. The point-slope equation is a useful tool for representing a line that passes through a given point with a specified slope.

The Given Point and Slope

In this problem, we are given a point ${(3, -2)}$ and a slope of ${-\frac{4}{5}}$. We need to find the point-slope equation that represents a line that passes through this point with the specified slope.

Analyzing the Options

Let's analyze the options given:

Option A: ${y - 3 = -\frac{4}{5}(x + 2)}$

In this option, the point ${(3, -2)}$ is not in the correct form. The point is given as ${(3, -2)}$, but in the equation, it is written as ${(x + 2)}$. This is incorrect because the point should be in the form of ${(x - x_1)}$.

Option B: ${y - 2 = \frac{4}{5}(x - 3)}$

In this option, the point ${(3, -2)}$ is in the correct form. The point is given as ${(3, -2)}$, and in the equation, it is written as ${(x - 3)}$. However, the slope is given as ${\frac{4}{5}}$, which is the negative reciprocal of the given slope.

Option C: ${y + 2 = -\frac{4}{5}(x - 3)}$

In this option, the point ${(3, -2)}$ is in the correct form. The point is given as ${(3, -2)}$, and in the equation, it is written as ${(x - 3)}$. The slope is also given as ${-\frac{4}{5}}$, which matches the given slope.

Based on the analysis of the options, we can conclude that the point-slope equation that represents a line that passes through the point ${(3, -2)}$ with a slope of ${-\frac{4}{5}}$ is:

${y + 2 = -\frac{4}{5}(x - 3)}$

This equation is in the correct form, with the point ${(3, -2)}$ in the form of ${(x - x_1)}$ and the slope given as ${-\frac{4}{5}}$.

The final answer is:

• Option C: ${y + 2 = -\frac{4}{5}(x - 3)}$
  Point-Slope Equation: A Comprehensive Guide with Q&A

In our previous article, we explored the concept of the point-slope equation and how it is used to represent a line that passes through a given point with a specified slope. In this article, we will delve deeper into the world of point-slope equations and answer some frequently asked questions.

Q1: What is the point-slope equation?

A1: The point-slope equation is a linear equation that takes the form of ${y - y_1 = m(x - x_1)}$, where ${m}$ is the slope of the line, and ${(x_1, y_1)}$ is the given point through which the line passes.

Q2: How do I write a point-slope equation?

A2: To write a point-slope equation, you need to know the slope of the line and a point through which the line passes. You can then use the formula ${y - y_1 = m(x - x_1)}$ to write the equation.

Q3: What is the significance of the point-slope equation?

A3: The point-slope equation is a useful tool for representing a line that passes through a given point with a specified slope. It is also used to find the equation of a line that passes through two points.

Q4: How do I find the slope of a line using the point-slope equation?

A4: To find the slope of a line using the point-slope equation, you can rearrange the equation to isolate the slope. For example, if the equation is ${y - 2 = 3(x - 1)}$, you can rearrange it to get ${y - 2 = 3x - 3}$, and then divide both sides by ${x - 1}$ to get ${m = 3}$.

Q5: Can I use the point-slope equation to find the equation of a line that passes through two points?

A5: Yes, you can use the point-slope equation to find the equation of a line that passes through two points. You can use the formula ${y - y_1 = m(x - x_1)}$ and substitute the coordinates of the two points to find the equation.

Q6: What is the difference between the point-slope equation and the slope-intercept equation?

A6: The point-slope equation and the slope-intercept equation are two different forms of linear equations. The point-slope equation takes the form of ${y - y_1 = m(x - x_1)}$, while the slope-intercept equation takes the form of ${y = mx + b}$.

Q7: Can I use the point-slope equation to find the equation of a line that is parallel to another line?

A7: Yes, you can use the point-slope equation to find the equation of a line that is parallel to another line. You can use the formula ${y - y_1 = m(x - x_1)}$ and substitute the slope of the parallel line to find the equation.

Q8: What is the significance of the slope in the point-slope equation?

A8: The slope in the point-slope equation represents the steepness of the line. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right.

Q9: Can I use the point-slope equation to find the equation of a line that is perpendicular to another line?

A9: Yes, you can use the point-slope equation to find the equation of a line that is perpendicular to another line. You can use the formula ${y - y_1 = m(x - x_1)}$ and substitute the negative reciprocal of the slope of the perpendicular line to find the equation.

Q10: What is the final answer to the problem of finding the point-slope equation that represents a line that passes through the point ${(3, -2)}$ with a slope of ${-\frac{4}{5}}$?

A10: The final answer is:

• Option C: ${y + 2 = -\frac{4}{5}(x - 3)}$