Find the value of k so that 48x-ky=11 and (k+2)x+16y=-19 are perpendicular lines.

k = -1 +/- √769

Step-by-step explanation:

48x - ky = 11

-48x           -48x

         -ky = -48x + 11

Slope:

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 (k + 2)x + 16y = -19

- (k + 2)x             -(k + 2)x

                 16y = -(k + 2)x - 19

Slope:

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and are perpendicular so they have opposite signs and are reciprocals of each other.  When multiplied by its reciprocal, their product equals -1.

*   = -1

= 1

Cross multiply, then solve for the variable.

(k + 2)(k) = 16(48)

k² + 2k - 768 = 0

Use quadratic formula to solve:

k = -1 +/- √769

.24

step-by-step explanation:

the probability is   .24

the first player had the highest probability of getting a silver marble

the game could be made fair by adding back the marble that was drawn after each draw