Find The Equation That Will Satisfy The Given Conditions.Find The Equation Of The Line That Contains The Point- (-16, -12) And Is Perpendicular To The Line 5x-3y+13=0- 2. Given The X-intercept Of - And Y-intercept Of -13.Find The Equation Of The Line

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Introduction

In mathematics, finding the equation of a line is a fundamental concept that involves using various techniques to determine the relationship between the x and y coordinates of points on the line. In this article, we will explore how to find the equation of a line that contains a given point and is perpendicular to another line. We will also discuss how to find the equation of a line given its x and y intercepts.

Finding the Equation of a Line with a Given Point and Perpendicular to Another Line

To find the equation of a line that contains a given point and is perpendicular to another line, we need to follow these steps:

  1. Find the slope of the given line: The slope of a line is a measure of how steep it is. It can be found by rearranging the equation of the line in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
  2. Find the slope of the perpendicular line: The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the given line.
  3. Use the point-slope form to find the equation of the line: The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Step 1: Find the Slope of the Given Line

The given line is 5x - 3y + 13 = 0. To find the slope, we need to rearrange the equation in the slope-intercept form. We can do this by isolating y:

y = (5/3)x + 13/3

The slope of the given line is 5/3.

Step 2: Find the Slope of the Perpendicular Line

The slope of the perpendicular line is the negative reciprocal of the slope of the given line. Therefore, the slope of the perpendicular line is -3/5.

Step 3: Use the Point-Slope Form to Find the Equation of the Line

We are given the point (-16, -12) that lies on the line. We can use the point-slope form to find the equation of the line:

y - (-12) = (-3/5)(x - (-16))

Simplifying the equation, we get:

y + 12 = (-3/5)(x + 16)

Multiplying both sides by 5, we get:

5y + 60 = -3x - 48

Rearranging the equation, we get:

3x + 5y + 108 = 0

Therefore, the equation of the line that contains the point (-16, -12) and is perpendicular to the line 5x - 3y + 13 = 0 is 3x + 5y + 108 = 0.

Finding the Equation of a Line with Given X and Y Intercepts

To find the equation of a line with given x and y intercepts, we can use the intercept form of a line, which is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.

Step 1: Write the Equation of the Line in the Intercept Form

We are given the x-intercept as - and the y-intercept as -13. We can write the equation of the line in the intercept form:

x/- + y/(-13) = 1

Simplifying the equation, we get:

-x/- + y/(-13) = 1

Multiplying both sides by -13, we get:

13x/- + y = -13

Rearranging the equation, we get:

13x + y = 0

Therefore, the equation of the line with x-intercept - and y-intercept -13 is 13x + y = 0.

Conclusion

In this article, we have discussed how to find the equation of a line with given conditions. We have shown how to find the equation of a line that contains a given point and is perpendicular to another line, and how to find the equation of a line with given x and y intercepts. By following these steps, we can determine the relationship between the x and y coordinates of points on a line and find the equation of the line.

References

Glossary

  • Slope: A measure of how steep a line is.
  • Negative Reciprocal: The negative reciprocal of a number is the number that, when multiplied by the original number, gives -1.
  • Point-Slope Form: A form of a line that uses the slope and a point on the line to find the equation of the line.
  • Intercept Form: A form of a line that uses the x and y intercepts to find the equation of the line.

Introduction

In our previous article, we discussed how to find the equation of a line with given conditions. We covered how to find the equation of a line that contains a given point and is perpendicular to another line, and how to find the equation of a line with given x and y intercepts. In this article, we will answer some frequently asked questions about finding the equation of a line.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It can be found by rearranging the equation of the line in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I find the slope of a line if it is not in slope-intercept form?

A: To find the slope of a line if it is not in slope-intercept form, you can rearrange the equation to isolate y. Once you have isolated y, you can compare it to the slope-intercept form to find the slope.

Q: What is the negative reciprocal of a number?

A: The negative reciprocal of a number is the number that, when multiplied by the original number, gives -1. For example, the negative reciprocal of 2 is -1/2, because 2 multiplied by -1/2 is -1.

Q: How do I find the equation of a line that contains a given point and is perpendicular to another line?

A: To find the equation of a line that contains a given point and is perpendicular to another line, you need to follow these steps:

  1. Find the slope of the given line.
  2. Find the slope of the perpendicular line, which is the negative reciprocal of the slope of the given line.
  3. Use the point-slope form to find the equation of the line.

Q: How do I find the equation of a line with given x and y intercepts?

A: To find the equation of a line with given x and y intercepts, you can use the intercept form of a line, which is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.

Q: What is the difference between the slope-intercept form and the intercept form of a line?

A: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. The intercept form of a line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. The slope-intercept form is used to find the equation of a line when you know the slope and a point on the line, while the intercept form is used to find the equation of a line when you know the x and y intercepts.

Q: Can I use the point-slope form to find the equation of a line with given x and y intercepts?

A: No, you cannot use the point-slope form to find the equation of a line with given x and y intercepts. The point-slope form is used to find the equation of a line when you know the slope and a point on the line, while the intercept form is used to find the equation of a line when you know the x and y intercepts.

Q: How do I know which form to use to find the equation of a line?

A: To determine which form to use to find the equation of a line, you need to consider the information you have about the line. If you know the slope and a point on the line, you can use the point-slope form. If you know the x and y intercepts, you can use the intercept form.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a line. We have discussed the slope-intercept form, the intercept form, and the point-slope form, and how to use them to find the equation of a line. By following these steps and using the correct form, you can determine the relationship between the x and y coordinates of points on a line and find the equation of the line.

References

Glossary

  • Slope: A measure of how steep a line is.
  • Negative Reciprocal: The negative reciprocal of a number is the number that, when multiplied by the original number, gives -1.
  • Point-Slope Form: A form of a line that uses the slope and a point on the line to find the equation of the line.
  • Intercept Form: A form of a line that uses the x and y intercepts to find the equation of the line.