Is there a clever way to attack this question from the SAT? It has to do with rational exponents and then you have to solve simultaneous equations: #a^3-ab^2=24# #a^2-ab=6#

Question

IN essence we need to solve this system of equations:

#a^3-ab^2=24#
#a^2-ab=6#

To solve them using brute force is long and nasty.

Since it's on the SAT, there's got to be a clever way to do this.

Any insight is helpful.

Thanks!

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Answer 1 · 2017-11-20T18:14:11

#(-1, 5)# and #(3, 1)#

#a^3-ab^2=24# and #a^2-ab=6#

#(a^3-ab^2)/(a^2-ab)=24/6#

#(a*(a^2-b^2))/[a*(a-b)]=4#

#(a^2-b^2)/(a-b)=4#

#((a+b)*(a-b))/(a-b)=4#

#a+b=4#, so #b=4-a#

Hence,

#a^2-a*(4-a)=6#

#a^2-4a+a^2=6#

#2a^2-4a-6=0#

#a^2-2a-3=0#

#(a+1)*(a-3)=0#, so #a_1=-1# and #a_2=3#

For #a=-1#, #b_1=5# and for #a=3#, #b_2=1#

Thus solutions of #(a, b)# are #(-1, 5)# and #(3, 1)#.