Understanding the Equation of the Line AB

Oct 13, 2025 by windailys.com 146 views

Understanding the Equation of the Line AB

The equation of the line AB is given as 3x - 2y = 6. This is a linear equation in the form of Ax + By = C, where A, B, and C are constants. To find the x-coordinate of point A, we need to find the point where the line intersects the y-axis. When the line intersects the y-axis, the x-coordinate is always 0.

Finding the x-coordinate of point A

To find the x-coordinate of point A, we substitute x = 0 into the equation 3x - 2y = 6.

3(0) - 2y = 6
-2y = 6
y = -3

So, the x-coordinate of point A is 0, and the y-coordinate is -3.

Finding the x-coordinate of point B

To find the x-coordinate of point B, we substitute y = 0 into the equation 3x - 2y = 6.

3x - 2(0) = 6
3x = 6
x = 2

So, the x-coordinate of point B is 2.

Finding the length of OA

The length of OA can be found using the distance formula, which is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In this case, the coordinates of point A are (0, -3), and the coordinates of point O are (0, 0). Substituting these values into the distance formula, we get:

d = \sqrt{(0 - 0)^2 + (-3 - 0)^2}
d = \sqrt{0 + 9}
d = \sqrt{9}
d = 3

So, the length of OA is 3.

Finding the length of OB

The length of OB can be found using the distance formula, which is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In this case, the coordinates of point B are (2, 0), and the coordinates of point O are (0, 0). Substituting these values into the distance formula, we get:

d = \sqrt{(2 - 0)^2 + (0 - 0)^2}
d = \sqrt{4 + 0}
d = \sqrt{4}
d = 2

So, the length of OB is 2.

Finding the x-coordinate and y-coordinate of point P

Since point P lies on the line AB, it must satisfy the equation 3x - 2y = 6. We can use this equation to find the x-coordinate and y-coordinate of point P.

Let's say the x-coordinate of point P is x. Then, the y-coordinate of point P is given by:

y = \frac{3x - 6}{2}

Since point P lies on the line AB, the x-coordinate of point P must be between 0 and 2. Let's say the x-coordinate of point P is x = 1.5. Then, the y-coordinate of point P is given by:

y = \frac{3(1.5) - 6}{2}
y = \frac{4.5 - 6}{2}
y = \frac{-1.5}{2}
y = -0.75

So, the x-coordinate and y-coordinate of point P are 1.5 and -0.75, respectively.

Conclusion

In this article, we have found the x-coordinate of point A, the length of OA, the length of OB, and the x-coordinate and y-coordinate of point P. We have used the equation of the line AB to find these values. The x-coordinate of point A is 0, the length of OA is 3, the length of OB is 2, and the x-coordinate and y-coordinate of point P are 1.5 and -0.75, respectively.

Frequently Asked Questions

Q: What is the equation of the line AB?

A: The equation of the line AB is given as 3x - 2y = 6.

Q: What is the x-coordinate of point A?

A: The x-coordinate of point A is 0.

Q: What is the y-coordinate of point A?

A: The y-coordinate of point A is -3.

Q: What is the x-coordinate of point B?

A: The x-coordinate of point B is 2.

Q: What is the length of OA?

A: The length of OA is 3.

Q: What is the length of OB?

A: The length of OB is 2.

Q: What is the x-coordinate and y-coordinate of point P?

A: The x-coordinate of point P is 1.5, and the y-coordinate of point P is -0.75.

Q: How did you find the x-coordinate and y-coordinate of point P?

A: We used the equation of the line AB to find the x-coordinate and y-coordinate of point P. We substituted x = 1.5 into the equation 3x - 2y = 6 to find the y-coordinate of point P.

Q: What is the significance of the equation 3x - 2y = 6?

A: The equation 3x - 2y = 6 represents the line AB. It is a linear equation in the form of Ax + By = C, where A, B, and C are constants.

Q: How did you find the length of OA and OB?

A: We used the distance formula to find the length of OA and OB. The distance formula is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

We substituted the coordinates of points A and O into the distance formula to find the length of OA, and we substituted the coordinates of points B and O into the distance formula to find the length of OB.

Q: What is the relationship between the x-coordinate and y-coordinate of point P?

A: The x-coordinate and y-coordinate of point P are related by the equation 3x - 2y = 6. We can substitute the x-coordinate of point P into this equation to find the y-coordinate of point P.

Q: How can I use the equation 3x - 2y = 6 to find the x-coordinate and y-coordinate of other points on the line AB?

A: You can substitute the x-coordinate of any point on the line AB into the equation 3x - 2y = 6 to find the y-coordinate of that point. Conversely, you can substitute the y-coordinate of any point on the line AB into the equation 3x - 2y = 6 to find the x-coordinate of that point.

Q: What is the significance of the line AB?

A: The line AB is a linear equation that represents a straight line on the coordinate plane. It has a slope of 3/2 and a y-intercept of -3.

Q: How can I graph the line AB?

A: You can graph the line AB by plotting the points A and B on the coordinate plane and drawing a straight line through them. Alternatively, you can use the equation 3x - 2y = 6 to graph the line AB.

Q: What is the relationship between the line AB and the point P?

A: The point P lies on the line AB, which means that it satisfies the equation 3x - 2y = 6. We can use this equation to find the x-coordinate and y-coordinate of point P.

Q: How can I use the equation 3x - 2y = 6 to find the x-coordinate and y-coordinate of other points on the line AB?

Q: What is the significance of the x-coordinate and y-coordinate of point P?

A: The x-coordinate and y-coordinate of point P are important because they represent the location of point P on the coordinate plane. We can use these coordinates to find the distance between point P and other points on the line AB.

Q: How can I use the x-coordinate and y-coordinate of point P to find the distance between point P and other points on the line AB?

A: You can use the distance formula to find the distance between point P and other points on the line AB. The distance formula is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

You can substitute the coordinates of point P and the other point into this formula to find the distance between them.