Question

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

Answer 1

M = 1/2

Step-by-step explanation:

To find the value of M such that the lines -2x + My = 5 and 4y + x = -9 are perpendicular, we need to determine the slopes of the lines. The slope of a line is given by the coefficient of x when the equation is in the form y = mx + b, where m is the slope.

First, let's transform the second given equation into slope-intercept form (y = mx + b):

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

Here, the slope of the second line is -1/4. For two lines to be perpendicular, their slopes must be negative reciprocals of each other. The negative reciprocal of -1/4 is 4. Thus the slope M for the first line must be 4 to make it perpendicular to the second line.

So, for the first equation -2x + My = 5, rearranging into slope-intercept form:

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

Comparing with y = mx + b, the slope m is 2/M. Therefore, to have a slope of 4, we set 2/M equal to 4:

2/M = 4
M = 2/4
M = 1/2

Answer 2

Answer: M = 1/2

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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.

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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.

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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