Which Is The Equation Of A Line That Has A Slope Of − 2 3 -\frac{2}{3} − 3 2 ​ And Passes Through The Point ( − 3 , − 1 (-3, -1 ( − 3 , − 1 ]?A. Y = − 2 3 X + 1 Y = -\frac{2}{3}x + 1 Y = − 3 2 ​ X + 1 B. Y = − 2 3 X + 3 Y = -\frac{2}{3}x + 3 Y = − 3 2 ​ X + 3 C. Y = − 2 3 X − 1 Y = -\frac{2}{3}x - 1 Y = − 3 2 ​ X − 1 D. $y =

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. The equation of a line can be expressed in various forms, including the slope-intercept form, point-slope form, and standard form. In this article, we will focus on finding the equation of a line that has a given slope and passes through a specific point.

Understanding the Slope

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter 'm' and is usually expressed as a fraction or a decimal. In this case, the slope of the line is given as $-\frac{2}{3}$.

Using the Point-Slope Form

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. We are given that the slope is $-\frac{2}{3}$ and the point is $(-3, -1)$. Plugging these values into the point-slope form, we get:

$$y - (-1) = -\frac{2}{3}(x - (-3))$$

Simplifying the equation, we get:

$$y + 1 = -\frac{2}{3}(x + 3)$$

Converting to Slope-Intercept Form

To convert the equation to slope-intercept form, we need to isolate the variable 'y' on one side of the equation. We can do this by multiplying both sides of the equation by $-\frac{3}{2}$:

$$-\frac{3}{2}(y + 1) = -\frac{2}{3}(x + 3)$$

Expanding and simplifying the equation, we get:

$$-\frac{3}{2}y - \frac{3}{2} = -\frac{2}{3}x - 2$$

Multiplying both sides of the equation by $-\frac{2}{3}$, we get:

$$y + 1 = \frac{2}{3}x + \frac{4}{3}$$

Subtracting 1 from both sides of the equation, we get:

$$y = \frac{2}{3}x + \frac{1}{3}$$

However, this is not one of the options. We need to go back to the point-slope form and try a different approach.

Using the Point-Slope Form Again

Let's go back to the point-slope form and try a different approach. We can rewrite the equation as:

$$y - (-1) = -\frac{2}{3}(x - (-3))$$

Simplifying the equation, we get:

$$y + 1 = -\frac{2}{3}(x + 3)$$

Multiplying both sides of the equation by $-3$, we get:

$$-3(y + 1) = 2(x + 3)$$

Expanding and simplifying the equation, we get:

$$-3y - 3 = 2x + 6$$

Adding 3 to both sides of the equation, we get:

$$-3y = 2x + 9$$

Dividing both sides of the equation by $-3$, we get:

$$y = -\frac{2}{3}x - 3$$

This is one of the options. Therefore, the correct equation of the line is:

$y = -\frac{2}{3}x - 3$

Conclusion

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Introduction

In our previous article, we discussed how to find the equation of a line that has a given slope and passes through a specific point. In this article, we will answer some frequently asked 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 two variables, usually x and y. It can be expressed in various forms, including the slope-intercept form, point-slope form, and standard 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 given by:

$$y = mx + b$$

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

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 given by:

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

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 two points?

A: To find the equation of a line that passes through two points, you can use the point-slope form. First, find the slope of the line using the formula:

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

Then, plug in the values of the two points and the slope into the point-slope form.

Q: How do I find the equation of a line that has a given slope and passes through a specific point?

A: To find the equation of a line that has a given slope and passes through a specific point, you can use the point-slope form. First, plug in the values of the point and the slope into the point-slope form. Then, simplify the equation to find the equation of the line.

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

A: The standard form of the equation of a line is given by:

$$Ax + By = C$$

where A, B, and C are constants.

Q: How do I convert the equation of a line from one form to another?

A: To convert the equation of a line from one form to another, you can use algebraic manipulations. For example, you can multiply both sides of the equation by a constant to eliminate a variable.

Q: What are some common mistakes to avoid when finding the equation of a line?

A: Some common mistakes to avoid when finding the equation of a line include:

• Not using the correct form of the equation
• Not plugging in the correct values into the equation
• Not simplifying the equation correctly
• Not checking the equation for errors

Conclusion

In this article, we have answered some frequently asked questions related to the equation of a line. We have discussed the different forms of the equation of a line, how to find the equation of a line that passes through two points, and how to convert the equation of a line from one form to another. We have also highlighted some common mistakes to avoid when finding the equation of a line.