Other Equations that Represent the Line

Oct 13, 2025 by miodeya.com 73 views

Introduction

In mathematics, a line is a fundamental concept that can be represented in various ways. One of the most common methods is by using the slope-intercept form, which is given by the equation y = mx + b, where m is the slope and b is the y-intercept. However, there are other forms of equations that can also represent a line, such as the point-slope form and the standard form. In this article, we will explore the equation of a line that passes through two given points and determine which other equations also represent the same line.

The Given Equation

The given equation of the line is:

y - 11 = \frac{-4}{3}(x + 7)

This equation represents a line that passes through the points (-7, 11) and (8, -9). To understand this equation, let's break it down into its components. The slope of the line is given by the coefficient of x, which is \frac{-4}{3}. The y-intercept is given by the constant term, which is 11.

Slope-Intercept Form

The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope and b is the y-intercept. To convert the given equation to slope-intercept form, we can add 11 to both sides of the equation:

y = \frac{-4}{3}(x + 7) + 11

Simplifying the equation, we get:

y = \frac{-4}{3}x - \frac{28}{3} + 11

Combining like terms, we get:

y = \frac{-4}{3}x + \frac{17}{3}

This is the slope-intercept form of the line.

Point-Slope Form

The point-slope form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. To convert the given equation to point-slope form, we can use the point (-7, 11) and the slope \frac{-4}{3}:

y - 11 = \frac{-4}{3}(x + 7)

This is the point-slope form of the line.

Standard Form

The standard form of a line is given by the equation Ax + By = C, where A, B, and C are constants. To convert the given equation to standard form, we can multiply both sides of the equation by 3 to eliminate the fraction:

3(y - 11) = -4(x + 7)

Expanding the equation, we get:

3y - 33 = -4x - 28

Adding 33 to both sides of the equation, we get:

3y = -4x + 5

This is the standard form of the line.

Other Equations that Represent the Line

Now that we have the slope-intercept form, point-slope form, and standard form of the line, we can determine which other equations also represent the same line. To do this, we can use the fact that the slope of the line is \frac{-4}{3} and the y-intercept is 11.

y = \frac{-4}{3}x + 11: This equation represents the same line as the given equation.
y - 11 = \frac{-4}{3}(x + 7): This equation represents the same line as the given equation.
y = \frac{-4}{3}x - \frac{28}{3} + 11: This equation represents the same line as the given equation.
y - 11 = \frac{-4}{3}(x - 8): This equation represents the same line as the given equation.
y = \frac{-4}{3}x + \frac{17}{3}: This equation represents the same line as the given equation.
3y - 33 = -4x - 28: This equation represents the same line as the given equation.
3y = -4x + 5: This equation represents the same line as the given equation.

Conclusion

$In conclusion, the equation y - 11 = \frac{-4}{3}(x + 7) represents a line that passes through the points (-7, 11) and (8, -9). We have also shown that the slope-intercept form, point-slope form, and standard form of the line are all equivalent to the given equation. Additionally, we have determined which other equations also represent the same line.<br/> **A Line in Mathematics: Understanding Equations and Representations - Q&A**$

Introduction

In our previous article, we explored the equation of a line that passes through two given points and determined which other equations also represent the same line. In this article, we will answer some frequently asked questions related to the topic.

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

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

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

A: To convert the given equation to slope-intercept form, you can add 11 to both sides of the equation:

y = \frac{-4}{3}(x + 7) + 11

Simplifying the equation, you get:

y = \frac{-4}{3}x - \frac{28}{3} + 11

Combining like terms, you get:

y = \frac{-4}{3}x + \frac{17}{3}

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

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

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

A: To convert the given equation to point-slope form, you can use the point (-7, 11) and the slope \frac{-4}{3}:

y - 11 = \frac{-4}{3}(x + 7)

Q: What is the standard form of a line?

A: The standard form of a line is given by the equation Ax + By = C, where A, B, and C are constants.

Q: How do I convert the given equation to standard form?

A: To convert the given equation to standard form, you can multiply both sides of the equation by 3 to eliminate the fraction:

3(y - 11) = -4(x + 7)

Expanding the equation, you get:

3y - 33 = -4x - 28

Adding 33 to both sides of the equation, you get:

3y = -4x + 5

Q: Which other equations represent the same line?

A: The following equations represent the same line:

y = \frac{-4}{3}x + 11
y - 11 = \frac{-4}{3}(x + 7)
y = \frac{-4}{3}x - \frac{28}{3} + 11
y - 11 = \frac{-4}{3}(x - 8)
y = \frac{-4}{3}x + \frac{17}{3}
3y - 33 = -4x - 28
3y = -4x + 5

Q: How do I determine which equations represent the same line?

A: To determine which equations represent the same line, you can use the fact that the slope of the line is \frac{-4}{3} and the y-intercept is 11.

Conclusion

In conclusion, we have answered some frequently asked questions related to the equation of a line that passes through two given points. We have also shown that the slope-intercept form, point-slope form, and standard form of the line are all equivalent to the given equation. Additionally, we have determined which other equations also represent the same line.