Write The Equation Of A Line In Standard Form That Has An { X$}$-intercept { (-P, 0)$}$ And A { Y$}$-intercept { (0, -R)$}$.A. { Px - Ry = -PR$}$B. { Px - Ry = PR$} C . \[ C. \[ C . \[ Rx + Py =
Understanding the Standard Form of a Line
The standard form of a line is a way to express the equation of a line in a specific format. It is often used in mathematics and engineering to describe the relationship between the x and y coordinates of points on a line. In this article, we will explore how to write the equation of a line in standard form given its x-intercept and y-intercept.
What are X-Intercept and Y-Intercept?
The x-intercept of a line is the point where the line intersects the x-axis. At this point, the y-coordinate is always 0. The y-intercept of a line is the point where the line intersects the y-axis. At this point, the x-coordinate is always 0.
Given X-Intercept (-P, 0) and Y-Intercept (0, -R)
We are given that the x-intercept is (-P, 0) and the y-intercept is (0, -R). This means that the line intersects the x-axis at the point (-P, 0) and the y-axis at the point (0, -R).
Writing the Equation of a Line in Standard Form
To write the equation of a line in standard form, we need to use the x-intercept and y-intercept to determine the coefficients of the equation. The standard form of a line is given by:
Ax + By = C
where A, B, and C are constants.
Using the X-Intercept (-P, 0)
We know that the x-intercept is (-P, 0). This means that when x = -P, y = 0. We can substitute these values into the equation to get:
A(-P) + B(0) = C
Simplifying the equation, we get:
-AP = C
Using the Y-Intercept (0, -R)
We know that the y-intercept is (0, -R). This means that when x = 0, y = -R. We can substitute these values into the equation to get:
A(0) + B(-R) = C
Simplifying the equation, we get:
-RB = C
Combining the Equations
We have two equations:
-AP = C
-RB = C
Since both equations are equal to C, we can set them equal to each other:
-AP = -RB
Dividing both sides by -1, we get:
AP = RB
Solving for A and B
We can solve for A and B by dividing both sides of the equation by P and R, respectively:
A = R/P
B = P/R
Substituting A and B into the Equation
We can substitute the values of A and B into the equation:
Ax + By = C
Substituting A = R/P and B = P/R, we get:
(R/P)x + (P/R)y = C
Multiplying both sides by PR, we get:
Rx + Py = C
Conclusion
In this article, we have shown how to write the equation of a line in standard form given its x-intercept and y-intercept. We have used the x-intercept (-P, 0) and y-intercept (0, -R) to determine the coefficients of the equation. The standard form of a line is given by:
Rx + Py = C
where R, P, and C are constants.
Answer
The correct answer is:
C. Rx + Py = C
Discussion
The equation of a line in standard form is a powerful tool in mathematics and engineering. It allows us to describe the relationship between the x and y coordinates of points on a line. By using the x-intercept and y-intercept, we can determine the coefficients of the equation and write it in standard form.
Example
Suppose we have a line with an x-intercept of (-2, 0) and a y-intercept of (0, -3). We can use the equation:
Rx + Py = C
to write the equation of the line. Substituting R = 3 and P = 2, we get:
3x + 2y = C
We can solve for C by substituting the x and y values of a point on the line. For example, if the point is (1, -1), we can substitute x = 1 and y = -1 into the equation to get:
3(1) + 2(-1) = C
Simplifying the equation, we get:
3 - 2 = C
C = 1
Therefore, the equation of the line is:
3x + 2y = 1
Final Answer
The final answer is:
Frequently Asked Questions
In this article, we will answer some frequently asked questions about the equation of a line in standard form.
Q: What is the standard form of a line?
A: The standard form of a line is a way to express the equation of a line in a specific format. It is often used in mathematics and engineering to describe the relationship between the x and y coordinates of points on a line.
Q: How do I write the equation of a line in standard form?
A: To write the equation of a line in standard form, you need to use the x-intercept and y-intercept to determine the coefficients of the equation. The standard form of a line is given by:
Ax + By = C
where A, B, and C are constants.
Q: What is the x-intercept and y-intercept?
A: The x-intercept of a line is the point where the line intersects the x-axis. At this point, the y-coordinate is always 0. The y-intercept of a line is the point where the line intersects the y-axis. At this point, the x-coordinate is always 0.
Q: How do I find the coefficients A, B, and C?
A: To find the coefficients A, B, and C, you need to use the x-intercept and y-intercept to determine the values of R and P. Then, you can substitute these values into the equation:
Rx + Py = C
Q: What is the equation of a line in standard form?
A: The equation of a line in standard form is given by:
Rx + Py = C
where R, P, and C are constants.
Q: How do I solve for C?
A: To solve for C, you need to substitute the x and y values of a point on the line into the equation:
Rx + Py = C
Then, you can simplify the equation to find the value of C.
Q: What is the final answer?
A: The final answer is:
C. Rx + Py = C
Common Mistakes
Here are some common mistakes to avoid when writing the equation of a line in standard form:
- Not using the x-intercept and y-intercept: Make sure to use the x-intercept and y-intercept to determine the coefficients of the equation.
- Not substituting the values of R and P: Make sure to substitute the values of R and P into the equation.
- Not simplifying the equation: Make sure to simplify the equation to find the value of C.
Conclusion
In this article, we have answered some frequently asked questions about the equation of a line in standard form. We have also discussed some common mistakes to avoid when writing the equation of a line in standard form. By following these tips, you can write the equation of a line in standard form with confidence.
Final Answer
The final answer is:
C. Rx + Py = C