Questions on Word Problems: Evaluation, Substitution answered by real tutors!

    (-x)²y - y³ = 105
       x²y - y³ = 105
     y(x² - y²) = 105
y(x - y)(x + y) = 105

So we have three factors y, x-y, and x+y having
product 105.

Since 105 = 1·3·5·7 either one of those factors is 1 
or none are.

If none of the three are 1 then

{y, x-y, x+y} = {3,5,7}.  

x+y must be the largest, 7 so

{y, x-y} = {3,5}.

The sum of the two elements of the left set must equal the 
sum of the two elements on the right set, so

y + x-y = 3+5
      x = 8

But that's impossible since the largest factor x+y is only 7 

So one of those factors must be 1.  Let's try y=1

y(x - y)(x + y) = 105
1(x - 1)(x + 1) = 105
         x² - 1 = 105
             x² = 106
              x = √106
which is not an integer.  So the first factor y cannot be 1.

The largest factor x+y cannot be 1, so the second factor x-y must be 1, 
x-y = 1, so we have y = x-1,

Substituting that:

      y(x - y)(x + y) = 105
  (x - 1)(1)(x + x-1) = 105
      (x - 1)(2x - 1) = 105
         2x² - 3x + 1 = 105
       2x² - 3x - 104 = 0
         (x-8)(2x+13) = 0

So x - 8 = 0,  2x + 13 = 0 
       x = 8        2x = -13
                     x =  (discard)

So we have x = 8, and since y = x-1 = 8-1 = 7

So the only solution with positive integers is x = 8, y = 7.

Checking:

           (-x)²y - y³ = 105
           (-8)²7 - 7³ = 105
            64·7 - 343 = 105
             448 - 343 = 105
                   105 = 105

Edwin

The distance the officer has to make up is .2 miles. The speed at which the officer makes up the distance is the difference between the cars' speeds. That's 60 - 50 = 10 MPH.

.2 = 10 * t

.2/10 = t

.02 hours = t = 1.2 minutes = 1 minute 12 seconds.

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T + F = 15 Equation 1

20T + 50F = 390 Equation 2

Eq 1: T = 15-F (plug that into equation 2)

20(15-F) + 50F = 390

300 - 20F + 50F = 390

30F = 90

F = 3

T = 15-F = 15-3 = 12

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