Two different buses start their routes at the same corner at 9 A. M. Buses from one route stop at the library every 9 minutes. Buses from the other route stop at the library every 12 minutes. In what number of minutes will both buses end up at the library?

In 36 minutes both buses  will end up at the library.

Step-by-step explanation:

Given: Buses from one route stop at the library every 9 minutes.

Buses from the other route stop at the library every 12 minutes.

The number of minutes in which both buses will end up at the library= LCM(9,12)

Now , 9= 3 x 3

12= 2 x 2 x 3

LCM(9,12) = 2 x 2 x 3 x 3 = 36

In 36 minutes both buses  will end up at the library.

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1