Use The Point-slope Formula To Write An Equation Of The Line That Passes Through \[$(0, -1)\$\] And \[$(4, 0)\$\]. Write The Answer In Slope-intercept Form, If Possible.The Equation Of The Line Is \[$\square\$\].

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Introduction

The point-slope formula is a powerful tool in mathematics that allows us to write the equation of a line that passes through two given points. In this article, we will explore how to use the point-slope formula to write the equation of a line that passes through the points {(0, -1)$}$ and {(4, 0)$}$. We will also discuss how to write the equation in slope-intercept form, if possible.

The Point-Slope Formula

The point-slope formula is given by:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

where {(x_1, y_1)$}$ is a point on the line and {m$}$ is the slope of the line. To use this formula, we need to find the slope of the line and one of the points.

Finding the Slope

To find the slope of the line, we can use the formula:

m=y2−y1x2−x1m = \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. In this case, we have the points {(0, -1)$}$ and {(4, 0)$}$. Plugging these values into the formula, we get:

m=0−(−1)4−0=14m = \frac{0 - (-1)}{4 - 0} = \frac{1}{4}

Using the Point-Slope Formula

Now that we have the slope, we can use the point-slope formula to write the equation of the line. We will use the point {(0, -1)$}$ and the slope {\frac{1}{4}$}$. Plugging these values into the formula, we get:

y−(−1)=14(x−0)y - (-1) = \frac{1}{4}(x - 0)

Simplifying this equation, we get:

y+1=14xy + 1 = \frac{1}{4}x

Writing the Equation in Slope-Intercept Form

To write the equation in slope-intercept form, we need to isolate the variable {y$. We can do this by subtracting [1} from both sides of the equation:

y=14x−1y = \frac{1}{4}x - 1

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

Conclusion

In this article, we used the point-slope formula to write the equation of a line that passes through the points {(0, -1)$}$ and {(4, 0)$}$. We found the slope of the line and used the point-slope formula to write the equation. We also wrote the equation in slope-intercept form, if possible. The equation of the line is {y = \frac{1}{4}x - 1$}$.

Example Problems

Problem 1

Use the point-slope formula to write the equation of a line that passes through the points {(2, 3)$}$ and {(4, 5)$}$.

Solution

To solve this problem, we need to find the slope of the line and one of the points. We can use the formula:

m=y2−y1x2−x1m = \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. In this case, we have the points {(2, 3)$}$ and {(4, 5)$}$. Plugging these values into the formula, we get:

m=5−34−2=22=1m = \frac{5 - 3}{4 - 2} = \frac{2}{2} = 1

Now that we have the slope, we can use the point-slope formula to write the equation of the line. We will use the point {(2, 3)$}$ and the slope ${$1$. Plugging these values into the formula, we get:

y−3=1(x−2)y - 3 = 1(x - 2)

Simplifying this equation, we get:

y−3=x−2y - 3 = x - 2

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

y=x+1y = x + 1

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

Problem 2

Use the point-slope formula to write the equation of a line that passes through the points {(0, 2)$}$ and {(3, 4)$}$.

Solution

To solve this problem, we need to find the slope of the line and one of the points. We can use the formula:

m=y2−y1x2−x1m = \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. In this case, we have the points {(0, 2)$}$ and {(3, 4)$}$. Plugging these values into the formula, we get:

m=4−23−0=23m = \frac{4 - 2}{3 - 0} = \frac{2}{3}

Now that we have the slope, we can use the point-slope formula to write the equation of the line. We will use the point {(0, 2)$}$ and the slope {\frac{2}{3}$. Plugging these values into the formula, we get:

y−2=23(x−0)y - 2 = \frac{2}{3}(x - 0)

Simplifying this equation, we get:

y−2=23xy - 2 = \frac{2}{3}x

Adding [2} to both sides of the equation, we get:

y=23x+2y = \frac{2}{3}x + 2

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

Tips and Tricks

  • When using the point-slope formula, make sure to use the correct slope and point.
  • When writing the equation in slope-intercept form, make sure to isolate the variable {y$.
  • When solving example problems, make sure to follow the same steps as in the solution.

Conclusion

In this article, we used the point-slope formula to write the equation of a line that passes through two given points. We found the slope of the line and used the point-slope formula to write the equation. We also wrote the equation in slope-intercept form, if possible. The equation of the line is [y = \frac{1}{4}x - 1\$}. We also solved two example problems and provided tips and tricks for using the point-slope formula.

Introduction

The point-slope formula is a powerful tool in mathematics that allows us to write the equation of a line that passes through two given points. In this article, we will answer some frequently asked questions about the point-slope formula.

Q: What is the point-slope formula?

A: The point-slope formula is a formula that allows us to write the equation of a line that passes through two given points. It is given by:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

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

Q: How do I find the slope of the line?

A: To find the slope of the line, you can use the formula:

m=y2−y1x2−x1m = \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.

Q: How do I use the point-slope formula to write the equation of a line?

A: To use the point-slope formula, you need to find the slope of the line and one of the points. Then, you can plug these values into the formula:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Simplifying this equation, you will get the equation of the line.

Q: Can I write the equation of a line in slope-intercept form using the point-slope formula?

A: Yes, you can write the equation of a line in slope-intercept form using the point-slope formula. To do this, you need to isolate the variable [$y$.

Q: What are some common mistakes to avoid when using the point-slope formula?

A: Some common mistakes to avoid when using the point-slope formula include:

  • Using the wrong slope or point
  • Not simplifying the equation correctly
  • Not isolating the variable [$y$]

Q: Can I use the point-slope formula to write the equation of a line that passes through three or more points?

A: No, the point-slope formula is only used to write the equation of a line that passes through two given points. If you have three or more points, you will need to use a different method to find the equation of the line.

Q: How do I know if the point-slope formula is the best method to use to write the equation of a line?

A: The point-slope formula is the best method to use to write the equation of a line when you have two points and you want to find the equation of the line that passes through those points.

Q: Can I use the point-slope formula to write the equation of a horizontal or vertical line?

A: No, the point-slope formula is not used to write the equation of a horizontal or vertical line. Instead, you can use the equation of a horizontal line, which is [y=c$,ortheequationofaverticalline,whichis\[y = c\$, or the equation of a vertical line, which is \[x = c$.

Conclusion

In this article, we answered some frequently asked questions about the point-slope formula. We hope that this article has been helpful in understanding how to use the point-slope formula to write the equation of a line that passes through two given points.

Additional Resources

  • For more information on the point-slope formula, see the article "Solving the Equation of a Line Using the Point-Slope Formula".
  • For more information on how to find the slope of a line, see the article "Finding the Slope of a Line".
  • For more information on how to write the equation of a line in slope-intercept form, see the article "Writing the Equation of a Line in Slope-Intercept Form".

Tips and Tricks

  • Make sure to use the correct slope and point when using the point-slope formula.
  • Make sure to simplify the equation correctly when using the point-slope formula.
  • Make sure to isolate the variable [$y$ when writing the equation of a line in slope-intercept form.

Conclusion

We hope that this article has been helpful in understanding how to use the point-slope formula to write the equation of a line that passes through two given points. If you have any further questions, please don't hesitate to ask.