A Line Passes Through The Point(-2,3) And Has A Slope Of 4. Write The Equation In Point Slope Form

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Introduction

In mathematics, the point-slope form of a linear equation is a powerful tool for describing lines on a coordinate plane. Given a point and a slope, we can use the point-slope form to write the equation of the line that passes through that point and has that slope. In this article, we will explore how to write the equation of a line in point-slope form, using the point (-2,3) and a slope of 4 as an example.

What is Point-Slope Form?

The point-slope form of a linear equation is a way of writing the equation of a line that passes through a given point and has a given slope. The general form of the point-slope equation is:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

How to Write the Equation in Point-Slope Form

To write the equation of a line in point-slope form, we need to know the coordinates of a point on the line and the slope of the line. In this case, we are given the point (-2,3) and the slope (m) is 4.

Step 1: Identify the Given Point and Slope

The given point is (-2,3) and the slope (m) is 4.

Step 2: Plug in the Values into the Point-Slope Equation

Now that we have the given point and slope, we can plug these values into the point-slope equation:

y - y1 = m(x - x1)

Substituting the given values, we get:

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

Step 3: Simplify the Equation

To simplify the equation, we need to distribute the slope (m) to the terms inside the parentheses:

y - 3 = 4x - 8

Step 4: Write the Equation in Point-Slope Form

The equation of the line in point-slope form is:

y - 3 = 4(x + 2)

Step 5: Simplify the Equation Further (Optional)

If we want to simplify the equation further, we can combine like terms:

y - 3 = 4x + 8

Conclusion

In this article, we have learned how to write the equation of a line in point-slope form using the point (-2,3) and a slope of 4 as an example. We have followed the steps to identify the given point and slope, plug in the values into the point-slope equation, simplify the equation, and write the equation in point-slope form. With this knowledge, you can now write the equation of a line in point-slope form using any given point and slope.

Example Problems

Problem 1

Write the equation of a line in point-slope form that passes through the point (1,2) and has a slope of 3.

Solution

y - 2 = 3(x - 1)

Problem 2

Write the equation of a line in point-slope form that passes through the point (-3,4) and has a slope of 2.

Solution

y - 4 = 2(x + 3)

Tips and Tricks

  • When writing the equation of a line in point-slope form, make sure to identify the given point and slope correctly.
  • Use the point-slope equation to write the equation of a line in point-slope form.
  • Simplify the equation by distributing the slope to the terms inside the parentheses.
  • Combine like terms to simplify the equation further (if necessary).

Conclusion

Q&A: Writing the Equation of a Line in Point-Slope Form

Frequently Asked Questions

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

A: The point-slope form of a linear equation is a way of writing the equation of a line that passes through a given point and has a given slope. The general form of the point-slope equation is:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Q: How do I write the equation of a line in point-slope form?

A: To write the equation of a line in point-slope form, you need to know the coordinates of a point on the line and the slope of the line. You can then plug these values into the point-slope equation and simplify the equation to get the final result.

Q: What if I don't have the slope of the line?

A: If you don't have the slope of the line, you can use the slope-intercept form of a linear equation, which is:

y = mx + b

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

Q: Can I use the point-slope form to find the slope of a line?

A: Yes, you can use the point-slope form to find the slope of a line. If you know the coordinates of two points on the line, you can use the point-slope equation to find the slope.

Q: How do I simplify the equation in point-slope form?

A: To simplify the equation in point-slope form, you can distribute the slope to the terms inside the parentheses and combine like terms.

Q: What if I have a negative slope?

A: If you have a negative slope, you can simply multiply the slope by -1 to get the positive slope.

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

A: No, you cannot use the point-slope form to write the equation of a vertical line. The point-slope form is only used for lines that are not vertical.

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

A: Yes, you can use the point-slope form to write the equation of a horizontal line. However, the slope of a horizontal line is always 0.

Q: How do I know if a line is vertical or horizontal?

A: A line is vertical if it has a slope of infinity (or undefined) and a horizontal line has a slope of 0.

Q: Can I use the point-slope form to write the equation of a line that passes through the origin?

A: Yes, you can use the point-slope form to write the equation of a line that passes through the origin. The point-slope equation is:

y - 0 = m(x - 0)

which simplifies to:

y = mx

Q: Can I use the point-slope form to write the equation of a line that passes through a point with a negative x-coordinate?

A: Yes, you can use the point-slope form to write the equation of a line that passes through a point with a negative x-coordinate. The point-slope equation is:

y - y1 = m(x - x1)

where (x1, y1) is the given point.

Q: Can I use the point-slope form to write the equation of a line that passes through a point with a negative y-coordinate?

A: Yes, you can use the point-slope form to write the equation of a line that passes through a point with a negative y-coordinate. The point-slope equation is:

y - y1 = m(x - x1)

where (x1, y1) is the given point.

Conclusion

In conclusion, the point-slope form of a linear equation is a powerful tool for describing lines on a coordinate plane. By following the steps outlined in this article, you can write the equation of a line in point-slope form using any given point and slope. With practice and patience, you will become proficient in writing the equation of a line in point-slope form.