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a · b = 20

b · c = 12

a + b + c =12

a=? b=? c=?

Aug 31, 2018

#1
+100516
+1

a · b = 20

b · c = 12

a + b + c =12

a=? b=? c=?

a =  20/b

c  = 12/ b

So

a +  b +  c  = 12    substitute for a, c

20/b + b + 12/b  =  12      multiply through by b

20 + b^2 + 12  = 12b     rearrange and simplify

b^2 - 12b + 32   =  0       factor

(b - 4) ( b - 8)  = 0

Set each factor to 0   and solve for  b  and we have that

b  = 4   or  b  = 8

If we want  a and c to  be integers, then b = 4  is the only solution

And  a  = 20 / b = 20 / 4  = 5

And c  = 12/b  = 12/4   = 3

So

a * b  = 5 * 4  = 20

b * c =  4 * 3   = 12

And

a + b + c  = 5 + 4 + 3   = 12

So  (a, b, c)  = ( 5 , 4, 3)

Aug 31, 2018
#2
+7598
+1

a · b  =  20      Divide both sides of this equation by  b  to solve for  a .

a  =  20 / b

b · c  =  12      Divide both sides of this equation by  b  to solve for  c .

c  =  12 / b

a + b + c  =  12

Substitute  20/b  in for  a ,  and substitute  12/b  in for  c .

$$\frac{20}{b}$$ + b + $$\frac{12}{b}$$  =  12

Multiply through by  b .

20 + b2 + 12  =  12b

Combine  20  and  12  to get  32 .

b2 + 32  =  12b

Subtract  12b  from both sides.

b2 - 12b + 32  =  0

Factor the left side.

(b - 4)(b - 8)  =  0

Set each factor equal to zero and solve for  b .

b - 4  =  0       or       b - 8  =  0

b  =  4         or         b  =  8

Now let's use these values of  b  to find  a  and  c .

If  b = 4  , then   a  =  20 / b  =  20 / 4  =  5

If  b = 4 ,  then   c  =  12 / b  =  12 / 4  =  3

So a solution is:   a  = 5 ,   b = 4 ,   and   c = 3

If  b = 8  , then   a  =  20 / b  =  20 / 8  =  2.5

If  b = 8 ,  then   c  =  12 / b  =  12 / 8  =  1.5

So a solution is:   a  = 2.5 ,   b = 8 ,   and   c = 1.5

Aug 31, 2018