Write the equation of a line that contains (-8,4) and the midpoint of the segment connecting (-10,5) and (-5,0) in standard form.
I'm not sure what I'm doing wrong. I found the midpoint (-7.5, -3.5), found the slope, and put the equation in standard form. I got x + y = -12, but the answer key tells me I'm wrong (the answer is 3x + y = -20). Can someone help me work it out?
midpoint of segment should be
(-7.5, 2.5)
Well, you found the midpoint wrong. It is (-7.5,2.5) unless there's a typo in the coordinates.
clearly x+y = -12 is wrong, since -8+4 = -4
That should have been your first clue.
Using correct midpoint, the slope is -3. So, the answer is
y-4 = -3(x+8)
y-4 = -3x-24
3x+y = -20
To find the equation of a line that contains the midpoint of the segment connecting two points, you first need to find the slope of the line passing through those two points. Then, you can use the slope-intercept form of a linear equation, y = mx + b, and substitute the midpoint coordinates to find the value of b.
Let's go step by step:
1. Find the midpoint of the segment connecting (-10, 5) and (-5, 0):
To find the midpoint, you can use the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Using (-10, 5) and (-5, 0), substitute the values into the formula:
Midpoint = ((-10 + (-5)) / 2, (5 + 0) / 2)
Midpoint = (-15 / 2, 5 / 2)
Midpoint = (-7.5, 2.5)
So the midpoint is (-7.5, 2.5).
2. Find the slope of the line passing through (-10, 5) and (-5, 0):
To find the slope, you can use the slope formula:
Slope = (y2 - y1) / (x2 - x1)
Using (-10, 5) and (-5, 0), substitute the values into the formula:
Slope = (0 - 5) / (-5 - (-10))
Slope = -5 / 5
Slope = -1
So the slope of the line passing through (-10, 5) and (-5, 0) is -1.
3. Use the slope-intercept form of a linear equation, y = mx + b, and substitute the midpoint coordinates (-7.5, 2.5) and the slope (-1) to find the value of b:
y = mx + b
2.5 = -1 * (-7.5) + b
2.5 = 7.5 + b
b = 2.5 - 7.5
b = -5
So the value of b is -5.
4. Write the equation in standard form:
To convert the equation to standard form, ax + by = c, you need to eliminate fractions (if any) and ensure that a, b, and c are integers.
In this case, start with the slope-intercept form: y = mx + b.
y = -x - 5
Now, multiply the entire equation by -1 to eliminate the negative sign in front of x:
-y = x + 5
Finally, rearrange the equation to have the x and y terms on one side and a constant term on the other side:
x + y = -5
So the equation of the line passing through the midpoint and (-8, 4) is x + y = -5 in standard form.
Therefore, the answer key is incorrect, and the equation you found, x + y = -12, is correct.