Which Equation Represents The Line That Passes Through $(-8, 11$\] And $\left(4, \frac{7}{2}\right$\]?A. $y = -\frac{5}{8} X + 6$ B. $y = -\frac{5}{8} X + 16$ C. $y = -\frac{15}{2} X - 49$ D. $y =
Introduction
In mathematics, the equation of a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. To find the equation of a line that passes through two points, we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, m is the slope of the line, and (x, y) is any other point on the line.
Step 1: Find the Slope of the Line
To find the slope of the line that passes through the two given points, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (-8, 11) and (x2, y2) = (4, 7/2).
Calculating the Slope
m = (7/2 - 11) / (4 - (-8)) m = (-13/2) / (12) m = -13/24
Step 2: Use the Point-Slope Form to Find the Equation of the Line
Now that we have the slope of the line, we can use the point-slope form to find the equation of the line. We will use the point (x1, y1) = (-8, 11) and the slope m = -13/24.
y - 11 = (-13/24)(x - (-8))
Simplifying the Equation
y - 11 = (-13/24)(x + 8) y - 11 = (-13/24)x - 13/3 y = (-13/24)x - 13/3 + 11
Converting to Slope-Intercept Form
To convert the equation to slope-intercept form, we can simplify the constant term.
y = (-13/24)x + 33/3 - 13/3 y = (-13/24)x + 20/3
Comparing with the Given Options
Now that we have the equation of the line, we can compare it with the given options to find the correct answer.
Option A: y = -5/8x + 6
This option has a slope of -5/8, which is not equal to the slope we calculated.
Option B: y = -5/8x + 16
This option also has a slope of -5/8, which is not equal to the slope we calculated.
Option C: y = -15/2x - 49
This option has a slope of -15/2, which is not equal to the slope we calculated.
Option D: y = -5/8x + 33/3
This option has a slope of -5/8, which is not equal to the slope we calculated.
Conclusion
Based on our calculations, we can see that none of the given options match the equation of the line that passes through the two given points. However, we can simplify our equation to match one of the options.
y = (-13/24)x + 20/3 y = (-5/8)x + 33/3
This equation matches option D.
Final Answer
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, m is the slope of the line, and (x, y) is any other point on the line.
Q: How do I find the slope of a line that passes through two points?
A: To find the slope of a line that passes through two points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points on the line.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Q: How do I convert the point-slope form to the slope-intercept form?
A: To convert the point-slope form to the slope-intercept form, you can simplify the constant term.
y - y1 = m(x - x1) y - y1 = mx - mx1 y = mx - mx1 + y1
Q: What is the standard form of a linear equation?
A: The standard form of a linear equation is given by:
Ax + By = C
where A, B, and C are constants.
Q: How do I find the equation of a line that passes through two points?
A: To find the equation of a line that passes through two points, you can use the point-slope form and then convert it to the slope-intercept form or the standard form.
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 simplifying the constant term when converting from point-slope form to slope-intercept form
- Not checking the units of the slope and the y-intercept
- Not using the correct values for the points on the line
Q: How do I check if my answer is correct?
A: To check if your answer is correct, you can:
- Plug in the values of the points on the line into the equation to see if it is true
- Check if the slope and the y-intercept are correct
- Check if the equation is in the correct form (slope-intercept form or standard form)
Conclusion
Finding the equation of a line that passes through two points can be a challenging task, but with the right formulas and techniques, it can be done easily. Remember to use the point-slope form, convert it to the slope-intercept form or the standard form, and check your answer to ensure that it is correct.