Which Equation Describes The Line Whose { X $}$-intercept Is { (4, 0)$}$ And Whose { Y $}$-intercept Is { (0, -3)$}$?A. { Y = 4x - 3 $}$
In mathematics, the equation of a line can be described in various forms, including slope-intercept form, point-slope form, and standard form. When given the x-intercept and y-intercept of a line, we can use this information to determine the equation of the line. In this article, we will explore how to find the equation of a line 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. Similarly, 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 Information: X-Intercept and Y-Intercept
We are given that the x-intercept of the line is (4, 0) and the y-intercept is (0, -3). This means that the line intersects the x-axis at the point (4, 0) and intersects the y-axis at the point (0, -3).
Using the Given Information to Find the Equation
To find the equation of the line, we can use the slope-intercept form of a line, which is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Since we are given the y-intercept, we can substitute this value into the equation:
y = mx + (-3)
Now, we need to find the slope of the line. We can use the x-intercept to find the slope. Since the x-intercept is (4, 0), we can substitute this point into the equation:
0 = m(4) + (-3)
Simplifying the equation, we get:
0 = 4m - 3
Adding 3 to both sides, we get:
3 = 4m
Dividing both sides by 4, we get:
m = 3/4
Now that we have the slope, we can substitute this value into the equation:
y = (3/4)x + (-3)
Simplifying the equation, we get:
y = (3/4)x - 3
Conclusion
In this article, we have explored how to find the equation of a line given its x-intercept and y-intercept. We used the slope-intercept form of a line and substituted the given values into the equation to find the slope and y-intercept. The final equation of the line is:
y = (3/4)x - 3
This equation describes the line whose x-intercept is (4, 0) and whose y-intercept is (0, -3).
Comparison with the Options
Let's compare our final equation with the options given:
A. y = 4x - 3
Our final equation is y = (3/4)x - 3, which is different from option A. Therefore, option A is not the correct answer.
Final Answer
The final answer is:
In the previous article, we explored how to find the equation of a line given its x-intercept and y-intercept. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the x-intercept of a line?
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.
Q: What is the y-intercept of a line?
A: 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 equation of a line given its x-intercept and y-intercept?
A: To find the equation of a line given its x-intercept and y-intercept, you can use the slope-intercept form of a line, which is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept. You can substitute the given values into the equation and solve for the slope and y-intercept.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: How do I find the slope of a line given its x-intercept and y-intercept?
A: To find the slope of a line given its x-intercept and y-intercept, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) is the x-intercept and (x2, y2) is the y-intercept. However, since the x-intercept has a y-coordinate of 0, you can simplify the formula to:
m = -b / x1
where b is the y-intercept and x1 is the x-coordinate of the x-intercept.
Q: What is the standard form of a line?
A: The standard form of a line is given by:
Ax + By = C
where A, B, and C are constants. This form is useful for finding the equation of a line given its x-intercept and y-intercept.
Q: How do I convert the slope-intercept form to the standard form?
A: To convert the slope-intercept form to the standard form, you can multiply both sides of the equation by the denominator of the slope. For example, if the slope-intercept form is:
y = (3/4)x - 3
You can multiply both sides by 4 to get:
4y = 3x - 12
Then, you can rearrange the equation to get:
3x - 4y = 12
This is the standard form of the line.
Q: What are some common mistakes to avoid when finding the equation of a line?
A: Some common mistakes to avoid when finding the equation of a line include:
- Not using the correct formula for the slope
- Not substituting the correct values into the equation
- Not simplifying the equation correctly
- Not checking the units of the variables
By avoiding these mistakes, you can ensure that you find the correct equation of the line.
Conclusion
In this article, we have answered some frequently asked questions related to finding the equation of a line given its x-intercept and y-intercept. We have covered topics such as the x-intercept and y-intercept, the slope-intercept form, and the standard form of a line. By following the steps outlined in this article, you can find the equation of a line with confidence.