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 = -\frac{4}{5}(x

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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 on a coordinate plane. It is a linear equation that takes the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. In this article, we will explore which point-slope equation represents a line that passes through a given point with a specified slope.

Understanding the Point-Slope Equation

The point-slope equation is a powerful tool used to represent a line on a coordinate plane. It is a linear equation that takes the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. The point-slope equation is a more general form of the slope-intercept equation, which is y = mx + b, where m is the slope and b is the y-intercept.

The Given Point and Slope

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

Analyzing the Options

Let's analyze the options given:

A. y - 3 = -4/5(x + 2) B. y - 2 = 4/5(x - 3) C. y + 2 = -4/5(x - 3)

We need to determine which of these options represents a line that passes through the point (3, -2) with a slope of -4/5.

Option A

Option A is y - 3 = -4/5(x + 2). To determine if this option represents a line that passes through the point (3, -2), we need to substitute the values of x and y into the equation.

y - 3 = -4/5(x + 2) -2 - 3 = -4/5(3 + 2) -5 = -4/5(5) -5 = -4

This equation is not true, so option A does not represent a line that passes through the point (3, -2) with a slope of -4/5.

Option B

Option B is y - 2 = 4/5(x - 3). To determine if this option represents a line that passes through the point (3, -2), we need to substitute the values of x and y into the equation.

y - 2 = 4/5(x - 3) -2 - 2 = 4/5(3 - 3) -4 = 4/5(0) -4 = 0

This equation is not true, so option B does not represent a line that passes through the point (3, -2) with a slope of -4/5.

Option C

Option C is y + 2 = -4/5(x - 3). To determine if this option represents a line that passes through the point (3, -2), we need to substitute the values of x and y into the equation.

y + 2 = -4/5(x - 3) -2 + 2 = -4/5(3 - 3) 0 = -4/5(0) 0 = 0

This equation is true, so option C represents a line that passes through the point (3, -2) with a slope of -4/5.

In conclusion, the point-slope equation that represents a line that passes through the point (3, -2) with a slope of -4/5 is option C: y + 2 = -4/5(x - 3). This equation is true and represents a line that passes through the given point with the specified slope.

Frequently Asked Questions About the Point-Slope Equation

The point-slope equation is a fundamental concept in mathematics used to represent a line on a coordinate plane. In this article, we will answer some frequently asked questions about the point-slope equation.

Q: What is the point-slope equation?

A: The point-slope equation is a linear equation that takes the form of y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.

Q: How do I use the point-slope equation to find the equation of a line?

A: To use the point-slope equation to find the equation of a line, you need to know the coordinates of a point on the line and the slope of the line. You can then substitute these values into the point-slope equation to find the equation of the line.

Q: What is the slope-intercept form of a line?

A: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I convert the point-slope equation to the slope-intercept form?

A: To convert the point-slope equation to the slope-intercept form, you need to isolate y on one side of the equation. You can do this by adding y1 to both sides of the equation and then subtracting m(x - x1) from both sides.

Q: What is the significance of the point-slope equation in real-world applications?

A: The point-slope equation has many real-world applications, including:

  • Physics: The point-slope equation is used to describe the motion of objects under the influence of gravity.
  • Engineering: The point-slope equation is used to design and optimize systems, such as bridges and buildings.
  • Computer Science: The point-slope equation is used in computer graphics and game development to create realistic and interactive 3D models.

Q: How do I graph a line using the point-slope equation?

A: To graph a line using the point-slope equation, you need to know the coordinates of a point on the line and the slope of the line. You can then use the point-slope equation to find the equation of the line and graph it on a coordinate plane.

Q: What are some common mistakes to avoid when using the point-slope equation?

A: Some common mistakes to avoid when using the point-slope equation include:

  • Incorrectly identifying the slope: Make sure to identify the slope correctly and use it in the point-slope equation.
  • Incorrectly identifying the point: Make sure to identify the point correctly and use it in the point-slope equation.
  • Not simplifying the equation: Make sure to simplify the equation to its simplest form.

In conclusion, the point-slope equation is a fundamental concept in mathematics used to represent a line on a coordinate plane. It has many real-world applications and is used in various fields, including physics, engineering, and computer science. By understanding the point-slope equation and its applications, you can solve problems and graph lines with ease.

  • Practice, practice, practice: The more you practice using the point-slope equation, the more comfortable you will become with it.
  • Use online resources: There are many online resources available that can help you learn and practice using the point-slope equation.
  • Seek help when needed: Don't be afraid to ask for help if you are struggling with the point-slope equation.