Complete The Equation Of The Line Through The Points { (2, 1)$}$ And { (5, -8)$}$. Use Exact Numbers.

Mar 10, 2025 by ADMIN 102 views

Complete the Equation of the Line Through Two Points

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Introduction

In mathematics, the equation of a line can be determined using various methods, including the slope-intercept form, point-slope form, and the two-point form. The two-point form is particularly useful when we are given two points on the line and need to find the equation of the line passing through them. In this article, we will use the two-point form to find the equation of the line passing through the points (2, 1) and (5, -8).

The Two-Point Form

The two-point form of a line is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) and (x2, y2) are two points on the line, and m is the slope of the line. To find the equation of the line passing through the points (2, 1) and (5, -8), we need to find the slope of the line first.

Finding the Slope

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the given points (2, 1) and (5, -8) into the formula, we get:

m = (-8 - 1) / (5 - 2) m = -9 / 3 m = -3

Writing the Equation of the Line

Now that we have found the slope of the line, we can use the two-point form to write the equation of the line passing through the points (2, 1) and (5, -8). We will use the point (2, 1) as (x1, y1) and the point (5, -8) as (x2, y2).

y - 1 = -3(x - 2)

Simplifying the Equation

To simplify the equation, we can expand the right-hand side and combine like terms:

y - 1 = -3x + 6 y = -3x + 7

Conclusion

In this article, we used the two-point form to find the equation of the line passing through the points (2, 1) and (5, -8). We first found the slope of the line using the formula m = (y2 - y1) / (x2 - x1), and then used the two-point form to write the equation of the line. The final equation of the line is y = -3x + 7.

Example Use Case

The equation of the line y = -3x + 7 can be used to find the y-coordinate of any point on the line, given its x-coordinate. For example, if we want to find the y-coordinate of the point with x-coordinate 4, we can substitute x = 4 into the equation:

y = -3(4) + 7 y = -12 + 7 y = -5

Therefore, the point with x-coordinate 4 has y-coordinate -5.

Applications of the Two-Point Form

The two-point form of a line has many applications in mathematics and science. Some of the most common applications include:

Graphing lines: The two-point form can be used to graph lines on a coordinate plane.
Finding the equation of a line: The two-point form can be used to find the equation of a line passing through two points.
Solving systems of equations: The two-point form can be used to solve systems of linear equations.
Modeling real-world problems: The two-point form can be used to model real-world problems, such as the motion of an object or the growth of a population.

Conclusion

In conclusion, the two-point form of a line is a powerful tool for finding the equation of a line passing through two points. By using the formula m = (y2 - y1) / (x2 - x1) to find the slope of the line, and then using the two-point form to write the equation of the line, we can find the equation of the line passing through any two points. The equation of the line can then be used to find the y-coordinate of any point on the line, given its x-coordinate.

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Q: What is the two-point form of a line?

A: The two-point form of a line is a mathematical formula used to find the equation of a line passing through two points. It is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) and (x2, y2) are two points on the line, and m is the slope of the line.

Q: How do I find the slope of a line using the two-point form?

A: To find the slope of a line using the two-point form, you can use the formula:

m = (y2 - y1) / (x2 - x1)

This formula calculates the slope of the line by dividing the difference in y-coordinates by the difference in x-coordinates.

Q: What if the two points have the same x-coordinate?

A: If the two points have the same x-coordinate, then the line passing through them is vertical. In this case, the equation of the line is simply x = x1, where x1 is the x-coordinate of the two points.

Q: What if the two points have the same y-coordinate?

A: If the two points have the same y-coordinate, then the line passing through them is horizontal. In this case, the equation of the line is simply y = y1, where y1 is the y-coordinate of the two points.

Q: Can I use the two-point form to find the equation of a line passing through three points?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have three points, you can use the three-point form of a line to find the equation of the line passing through them.

Q: How do I use the two-point form to solve a system of linear equations?

A: To use the two-point form to solve a system of linear equations, you can first find the equation of the line passing through the two points. Then, you can substitute the equation of the line into one of the equations in the system to solve for the variables.

Q: What are some real-world applications of the two-point form?

A: The two-point form has many real-world applications, including:

Graphing lines: The two-point form can be used to graph lines on a coordinate plane.
Finding the equation of a line: The two-point form can be used to find the equation of a line passing through two points.
Solving systems of equations: The two-point form can be used to solve systems of linear equations.
Modeling real-world problems: The two-point form can be used to model real-world problems, such as the motion of an object or the growth of a population.

Q: Can I use the two-point form to find the equation of a line passing through a point and a line?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have a point and a line, you can use the point-line form of a line to find the equation of the line passing through them.

Q: How do I use the two-point form to find the equation of a line passing through a point and a circle?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have a point and a circle, you can use the point-circle form of a line to find the equation of the line passing through them.

Q: Can I use the two-point form to find the equation of a line passing through a point and a parabola?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have a point and a parabola, you can use the point-parabola form of a line to find the equation of the line passing through them.

Q: How do I use the two-point form to find the equation of a line passing through a point and a hyperbola?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have a point and a hyperbola, you can use the point-hyperbola form of a line to find the equation of the line passing through them.

Q: Can I use the two-point form to find the equation of a line passing through a point and an ellipse?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have a point and an ellipse, you can use the point-ellipse form of a line to find the equation of the line passing through them.

Q: How do I use the two-point form to find the equation of a line passing through a point and a circle?

Q: Can I use the two-point form to find the equation of a line passing through a point and a parabola?

Q: How do I use the two-point form to find the equation of a line passing through a point and a hyperbola?

Q: Can I use the two-point form to find the equation of a line passing through a point and an ellipse?

Q: What are some common mistakes to avoid when using the two-point form?

A: Some common mistakes to avoid when using the two-point form include:

Not checking if the two points are on the same line: If the two points are on the same line, the two-point form will not work.
Not checking if the two points are on the same plane: If the two points are not on the same plane, the two-point form will not work.
Not checking if the two points are on the same line segment: If the two points are not on the same line segment, the two-point form will not work.
Not checking if the two points are on the same circle: If the two points are not on the same circle, the two-point form will not work.
Not checking if the two points are on the same parabola: If the two points are not on the same parabola, the two-point form will not work.
Not checking if the two points are on the same hyperbola: If the two points are not on the same hyperbola, the two-point form will not work.
Not checking if the two points are on the same ellipse: If the two points are not on the same ellipse, the two-point form will not work.

Q: How do I troubleshoot common issues with the two-point form?

A: To troubleshoot common issues with the two-point form, you can try the following:

Check if the two points are on the same line: If the two points are not on the same line, the two-point form will not work.
Check if the two points are on the same plane: If the two points are not on the same plane, the two-point form will not work.
Check if the two points are on the same line segment: If the two points are not on the same line segment, the two-point form will not work.
Check if the two points are on the same circle: If the two points are not on the same circle, the two-point form will not work.
Check if the two points are on the same parabola: If the two points are not on the same parabola, the two-point form will not work.
Check if the two points are on the same hyperbola: If the two points are not on the same hyperbola, the two-point form will not work.
Check if the two points are on the same ellipse: If the two points are not on the same ellipse, the two-point form will not work.

Q: Can I use the two-point form to find the equation of a line passing through a point and a line segment?

A: No, the two-point form can only be used to find the equation of a line passing through two points. If you have a point and a line segment, you can use the point-line segment form of a line to find the equation of the line passing through them.

Q: How do I use the two-point form to find the equation of a line passing through a point and a circle?

A: No, the