A Line Passes Through The Point { (4,1)$}$ And Has A Slope Of { \frac{5}{4}$}$.Write An Equation In Slope-intercept Form For This Line.

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Introduction

In this article, we will explore how to write an equation in slope-intercept form for a line that passes through a given point and has a known slope. 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. We will use the point-slope form of a line, which 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 of a Line

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. This form is useful when we know a point on the line and the slope of the line. We can use this form to write an equation for the line and then convert it to slope-intercept form.

Given Information

We are given that a line passes through the point (4, 1) and has a slope of 5/4. We can use this information to write an equation for the line in point-slope form.

Writing the Equation in Point-Slope Form

Using the point-slope form of a line, we can write an equation for the line as follows:

y - 1 = (5/4)(x - 4)

This equation is in point-slope form, where (4, 1) is the given point and 5/4 is the slope of the line.

Converting to Slope-Intercept Form

To convert the equation to slope-intercept form, we need to simplify the equation and isolate y. We can do this by distributing the slope to the terms inside the parentheses and then adding 1 to both sides of the equation.

y - 1 = (5/4)(x - 4) y - 1 = (5/4)x - 5 y = (5/4)x - 5 + 1 y = (5/4)x - 4

This equation is now in slope-intercept form, where 5/4 is the slope of the line and -4 is the y-intercept.

Conclusion

In this article, we have shown how to write an equation in slope-intercept form for a line that passes through a given point and has a known slope. We used the point-slope form of a line to write an equation for the line and then converted it to slope-intercept form. This process is useful when we know a point on the line and the slope of the line, and we need to write an equation for the line in slope-intercept form.

Example Problems

  1. A line passes through the point (2, 3) and has a slope of 2/3. Write an equation for the line in slope-intercept form.
  2. A line passes through the point (1, 2) and has a slope of 3/2. Write an equation for the line in slope-intercept form.
  3. A line passes through the point (3, 4) and has a slope of 1/2. Write an equation for the line in slope-intercept form.

Solutions

  1. Using the point-slope form of a line, we can write an equation for the line as follows:

y - 3 = (2/3)(x - 2)

Converting to slope-intercept form, we get:

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

  1. Using the point-slope form of a line, we can write an equation for the line as follows:

y - 2 = (3/2)(x - 1)

Converting to slope-intercept form, we get:

y = (3/2)x + 1/2

  1. Using the point-slope form of a line, we can write an equation for the line as follows:

y - 4 = (1/2)(x - 3)

Converting to slope-intercept form, we get:

y = (1/2)x - 1/2

Final Thoughts

Introduction

In our previous article, we explored how to write an equation in slope-intercept form for a line that passes through a given point and has a known slope. We used the point-slope form of a line to write an equation for the line and then converted it to slope-intercept form. In this article, we will answer some frequently asked questions about writing an equation in slope-intercept form for a line with a given point and slope.

Q&A

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 write an equation for a line in point-slope form?

A: To write an equation for a line in point-slope form, you need to know a point on the line and the slope of the line. You can use the point-slope form of a line, which 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 an equation from point-slope form to slope-intercept form?

A: To convert an equation from point-slope form to slope-intercept form, you need to simplify the equation and isolate y. You can do this by distributing the slope to the terms inside the parentheses and then adding the constant term to both sides of the equation.

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

A: To find the y-intercept of a line, you need to know the slope of the line and a point on the line. You can use the point-slope form of a line to write an equation for the line and then convert it to slope-intercept form. The y-intercept is the constant term in the slope-intercept form of the equation.

Q: What is the difference between the point-slope form and the slope-intercept 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. 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. The point-slope form is useful when you know a point on the line and the slope of the line, while the slope-intercept form is useful when you know the slope of the line and the y-intercept.

Q: Can I use the point-slope form to write an equation for a line with a given slope and y-intercept?

A: No, you cannot use the point-slope form to write an equation for a line with a given slope and y-intercept. The point-slope form requires you to know a point on the line, while the slope-intercept form requires you to know the slope of the line and the y-intercept.

Q: Can I use the slope-intercept form to write an equation for a line with a given point and slope?

A: Yes, you can use the slope-intercept form to write an equation for a line with a given point and slope. You can use the point-slope form to write an equation for the line and then convert it to slope-intercept form.

Example Problems

  1. A line passes through the point (2, 3) and has a slope of 2/3. Write an equation for the line in slope-intercept form.
  2. A line passes through the point (1, 2) and has a slope of 3/2. Write an equation for the line in slope-intercept form.
  3. A line passes through the point (3, 4) and has a slope of 1/2. Write an equation for the line in slope-intercept form.

Solutions

  1. Using the point-slope form of a line, we can write an equation for the line as follows:

y - 3 = (2/3)(x - 2)

Converting to slope-intercept form, we get:

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

  1. Using the point-slope form of a line, we can write an equation for the line as follows:

y - 2 = (3/2)(x - 1)

Converting to slope-intercept form, we get:

y = (3/2)x + 1/2

  1. Using the point-slope form of a line, we can write an equation for the line as follows:

y - 4 = (1/2)(x - 3)

Converting to slope-intercept form, we get:

y = (1/2)x - 1/2

Final Thoughts

In this article, we have answered some frequently asked questions about writing an equation in slope-intercept form for a line with a given point and slope. We have shown how to use the point-slope form of a line to write an equation for the line and then convert it to slope-intercept form. This process is useful when you know a point on the line and the slope of the line, and you need to write an equation for the line in slope-intercept form.