Select The Correct Answer.The Lines A B ↔ \overleftrightarrow{AB} A B And B C ↔ \overleftrightarrow{BC} BC Form A Right Angle At Point B B B . If A = ( − 3 , − 1 A=(-3,-1 A = ( − 3 , − 1 ] And B = ( 4 , 4 B=(4,4 B = ( 4 , 4 ], What Is The Equation Of

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. When given two points that form a right angle, we can use this information to find the equation of the line that passes through these points. In this article, we will explore how to find the equation of a line given two points that form a right angle.

Understanding the Problem

We are given two points, $A=(-3,-1)$ and $B=(4,4)$, that form a right angle at point $B$. Our goal is to find the equation of the line that passes through these two points. To do this, we need to understand the concept of slope and how it relates to the equation of a line.

Slope of a Line

The slope of a line is a measure of how steep it is. It is calculated by dividing the change in the y-coordinate by the change in the x-coordinate. Mathematically, the slope of a line is given by:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Finding the Slope

Using the given points $A=(-3,-1)$ and $B=(4,4)$, we can calculate the slope of the line that passes through these points.

$$m = \frac{4 - (-1)}{4 - (-3)} = \frac{5}{7}$$

Equation of a Line

Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation of the line that passes through the given points.

The point-slope form of the equation of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Finding the Equation

Using the point $B=(4,4)$ and the slope $m=\frac{5}{7}$, we can plug these values into the point-slope form of the equation of a line.

$$y - 4 = \frac{5}{7}(x - 4)$$

Simplifying the Equation

To simplify the equation, we can multiply both sides by 7 to eliminate the fraction.

$$7(y - 4) = 5(x - 4)$$

Expanding and simplifying the equation, we get:

$$7y - 28 = 5x - 20$$

Rearranging the terms, we get:

$$7y = 5x + 8$$

Conclusion

In this article, we have shown how to find the equation of a line given two points that form a right angle. We used the concept of slope and the point-slope form of the equation of a line to find the equation of the line that passes through the given points. The final equation of the line is:

$$7y = 5x + 8$$

This equation represents the relationship between the x and y coordinates of any point on the line.

Final Answer

The final answer is: $\boxed{7y = 5x + 8}$

Introduction

In our previous article, we explored how to find the equation of a line given two points that form a right angle. In this article, we will answer some common questions related to the equation of a line.

Q: What is the equation of a line?

A: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of any point on the line.

Q: How do I find the equation of a line given two points?

A: To find the equation of a line given two points, you need to calculate the slope of the line using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Then, use the point-slope form of the equation of a line:

$$y - y_1 = m(x - x_1)$$

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

A: The point-slope form of the equation of a line is:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line 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 by a common factor to eliminate any fractions. You can also rearrange the terms to put the equation in a more convenient form.

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

A: The slope-intercept form of the equation of a line is:

$$y = mx + b$$

where $m$ is the slope and $b$ is the y-intercept.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can set $x = 0$ in the equation of the line and solve for $y$.

Q: What is the equation of a line in standard form?

A: The equation of a line in standard form is:

$$Ax + By = C$$

where $A$, $B$, and $C$ are constants.

Q: How do I convert the equation of a line from slope-intercept form to standard form?

A: To convert the equation of a line from slope-intercept form to standard form, you can multiply both sides of the equation by the denominator of the fraction.

Q: What is the equation of a line in parametric form?

A: The equation of a line in parametric form is:

$$x = x_1 + at$$

$$y = y_1 + bt$$

where $(x_1, y_1)$ is a point on the line and $a$ and $b$ are constants.

Conclusion

In this article, we have answered some common questions related to the equation of a line. We have covered topics such as finding the equation of a line given two points, simplifying the equation of a line, and converting the equation of a line from slope-intercept form to standard form.

Final Answer

The final answer is: There is no single final answer to this article, as it is a collection of questions and answers related to the equation of a line.