Find m so that the lines with equations -2x + my = 5 and 4y + x = -9 are
perpendicular.

Work Shown:

Solve the second equation for y to get the following

4y+x = -9

4y = -x-9

y = (-x-9)/4

y = (-x)/4-9/4

y = (-1/4)x-9/4

The equation is in y = mx+b form where m = -1/4 is the slope of this line. Note: this is not the same as the uppercase M we're after; the lowercase m is conventionally used as the general slope term.

The slope of the second line is -1/4. Flip the fraction to get -4/1 = -4, then flip the sign to go from -4 to +4 or just 4.

This means the slope of the first line must be 4 in order for the two lines to be perpendicular. The two perpenicular slopes multiply to -1. We see that (-1/4)*(4) = -1.

Let's solve the first equation for y

-2x + My = 5

My = 2x+5

y = (2x+5)/M

y = (2x)/M + 5/M

y = (2/M)x + 5/M

The slope here is 2/M. It must be equal to 4, so lets equate and solve for M

2/M = 4

2 = 4M

4M = 2

M = 2/4

M = 1/2

which is our final answer