Answer:
D. 2x - y = 13
Explanation:
We are given the line x + 2y =5
We want to find the equation that is perpendicular to this line, and that also passes through (4, -5)
Perpendicular lines have slopes that multiply to -1
First, let's find the slope of the line
The line is currently in standard form (ax+by=c), which doesn't show the slope; we can convert this line into slope-intercept form (y=mx+b, where m is the slope and b is the y intercept)
To do that, we will need to solve the equation for y
Here it is:
x + 2y = 5
Subtract x from both sides
2y = -x + 5
Divide both sides by 2
y = -1/2x + 5/2
The slope of the line is -1/2
Now, to find the slope of the line perpendicular to it, use this equation:
-1/2m = -1
Multiply both sides by -2
m = 2
The slope of the line perpendicular to it is 2
Here is the equation of the line so far in slope-intercept form
y = 2x + b
We need to find b
As the equation passes through the point (4, -5), we can use it to help solve for b.
Substitute 4 as x and -5 as y.
-5 = 2(4) + b
Multiply
-5 = 8 + b
Subtract 8 from both sides
-13 = b
This is our line:
y = 2x - 13
Even though this is in slope-intercept form, all of the options in your question are in standard form. Ergo, we must convert the line into standard form.
To do this, we can subtract 2x from both sides, as in standard form, x and y are on the same side
-2x + y = -13
Now multiply both sides by -1
2x - y = 13
This makes the answer D.