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

When two lines are perpendicular, the product of their slopes are equal to -1.
The equation -2x + My = 5 written in slope-intercept form is
Y = \frac{2x}{M} +\frac{5}{M}
The slope of this is \frac{2}{M}
Similarly the equation 4y + x = -9 written in slope-intercept form is
y = \frac{-x}{4} -\frac{9}{4}
The slope of this is \frac{-1}{4}
The product of these slopes is equal to -1.
so,
\frac{2}{M}\ \times\frac{-1}{4}\ =-1
Solve for M to get,
M = \frac{1}{2}
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