Consider the following three systems of linear equations. System A System B System C 13x = 39 [81] x=3 [C1] - 5x – 4y = 9 [A1] 9x +2y = 15 [A2] 9x + 2y = 15 [B2] 9x +2y = 15 (C2] Answer the questions below. For each, choose the transformation and then fill in the blank with the correct number. The arrow (-) means the expression on the left becomes the expression on the right. How do we transform System A into System B?

hello from mrbilldoesmath!

answer:   (7x + 6y) ( 49x^2 - 42xy + 36y^2)

discussion:

in general, the factorization of a^3 + b^3 is

a^3 + b^ 3 = (a + b) ( a^2 - ab + b^2)

in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:

343x^3 + 216y^3 = (7x)^3 + (6y)^3

if you think of a (above) as 7x and b as 6y the factorization becomes

343x^3 + 216y^3 =

((7x + 6y) (   (7x)^2 - 7x*6y + (6y)^2)   =

(7x + 6y) ( 49x^2 - 42xy + 36y^2)

you,

mrb

(a + b) (a^2 - a b + b^2)