The Slope-intercept Form Of The Equation Of A Line That Passes Through The Point \[$(-2, -13)\$\] Is \[$y = 5x - 3\$\]. What Is The Point-slope Form Of The Equation For This Line?A. \[$y - 13 = 5(x - 2)\$\] B. \[$y + 13 =

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Introduction

In mathematics, the slope-intercept form and point-slope form are two ways to represent the equation of a line. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. On the other hand, the point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In this article, we will explore the relationship between the slope-intercept form and point-slope form of a line.

The Slope-Intercept Form

The slope-intercept form of the equation of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The slope-intercept form is a convenient way to represent the equation of a line, as it allows us to easily identify the slope and y-intercept of the line.

The Point-Slope Form

The point-slope form of the equation 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. The point-slope form is a more general form of the equation of a line, as it allows us to represent the equation of a line in terms of a point on the line and the slope of the line.

Converting from Slope-Intercept Form to Point-Slope Form

To convert from the slope-intercept form to the point-slope form, we need to use the point-slope form equation and substitute the values of the slope and the point on the line. Let's consider the given equation y = 5x - 3. We can see that the slope of the line is 5 and the y-intercept is -3.

Step 1: Identify the Slope and the Point on the Line

The slope of the line is 5 and the y-intercept is -3. We can use the point (-2, -13) as the point on the line.

Step 2: Substitute the Values into the Point-Slope Form Equation

We can substitute the values of the slope and the point on the line into the point-slope form equation:

y - y1 = m(x - x1)

Substituting the values, we get:

y - (-13) = 5(x - (-2))

Simplifying the equation, we get:

y + 13 = 5(x + 2)

Step 3: Simplify the Equation

We can simplify the equation by distributing the slope to the terms inside the parentheses:

y + 13 = 5x + 10

Conclusion

In conclusion, we have successfully converted the slope-intercept form of the equation of a line to the point-slope form. The point-slope form of the equation is y + 13 = 5(x + 2). This equation represents the same line as the original slope-intercept form equation y = 5x - 3.

Answer

The point-slope form of the equation for this line is:

y + 13 = 5(x + 2)

This is option B.

Discussion

The point-slope form of the equation of a line is a more general form of the equation of a line, as it allows us to represent the equation of a line in terms of a point on the line and the slope of the line. The point-slope form is useful when we need to find the equation of a line that passes through a given point and has a given slope.

Example

Let's consider another example. Suppose we have a line with a slope of 2 and a y-intercept of 4. We can use the point-slope form equation to find the equation of the line. Let's use the point (1, 6) as the point on the line.

y - y1 = m(x - x1)

Substituting the values, we get:

y - 6 = 2(x - 1)

Simplifying the equation, we get:

y - 6 = 2x - 2

Adding 6 to both sides, we get:

y = 2x + 4

This is the slope-intercept form of the equation of the line.

Conclusion

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 of the line and b is the y-intercept.

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 from slope-intercept form to point-slope form?

A: To convert from slope-intercept form to point-slope form, you need to use the point-slope form equation and substitute the values of the slope and the point on the line.

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

A: To find the slope and y-intercept of a line, you need to use the slope-intercept form equation y = mx + b. The slope (m) is the coefficient of x, and the y-intercept (b) is the constant term.

Q: What is the difference between slope-intercept form and point-slope form?

A: The main difference between slope-intercept form and point-slope form is that slope-intercept form gives the equation of a line in terms of the slope and y-intercept, while point-slope form gives the equation of a line in terms of a point on the line and the slope.

Q: When should I use slope-intercept form and when should I use point-slope form?

A: You should use slope-intercept form when you know the slope and y-intercept of a line, and you want to find the equation of the line. You should use point-slope form when you know a point on the line and the slope, and you want to find the equation of the line.

Q: Can I convert from point-slope form to slope-intercept form?

A: Yes, you can convert from point-slope form to slope-intercept form by using the point-slope form equation and substituting the values of the slope and the point on the line.

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 need to use the point-slope form equation and substitute the values of the two points and the slope.

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

A: The point-slope form is significant in real-world applications because it allows us to represent the equation of a line in terms of a point on the line and the slope, which is useful in many fields such as physics, engineering, and economics.

Q: Can I use point-slope form to find the equation of a line that is not a straight line?

A: No, you cannot use point-slope form to find the equation of a line that is not a straight line. Point-slope form is only used to find the equation of a straight line.

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

A: To graph a line using point-slope form, you need to use the point-slope form equation and substitute the values of the slope and the point on the line. Then, you can plot the point on the line and use the slope to draw the line.

Q: Can I use point-slope form to find the equation of a line that is parallel or perpendicular to another line?

A: Yes, you can use point-slope form to find the equation of a line that is parallel or perpendicular to another line. You need to use the point-slope form equation and substitute the values of the slope and the point on the line, and then use the properties of parallel and perpendicular lines to find the equation of the line.