Answer:
a = 2, b = 3.5
Step-by-step explanation:
Expanding [tex](ax+by)^7[/tex] using Binomial expansion, we have that:
[tex](ax+by)^7[/tex] =
[tex](ax)^7(by)^0 + (ax)^6(by)^1 + (ax)^5(by)^2 + (ax)^4(by)^3 + (ax)^3(by)^4 + (ax)^2(by)^5 + (ax)^1(by)^6 + (ax)^0(by)^7[/tex]
[tex]= (a)^7(x)^7+ (a)^6(x)^6(b)(y) + (a)^5(x)^5(b)^2(y)^2 + (a)^4(x)^4(b)^3(y)^3 + (a)^3(x)^3(b)^4(y)^4 + (a)^2(x)^2(b)^5(y)^5 + (a)(x)(b)^6(y)^6 + (b)^7(y)^7\\\\\\= (a)^7(x)^7+ (a)^6(b)(x)^6(y) + (a)^5(b)^2(x)^5(y)^2 + (a)^4(b)^3(x)^4(y)^3 + (a)^3(b)^4(x)^3(y)^4 + (a)^2(b)^5(x)^2(y)^5 + (a)(b)^6(x)(y)^6 + (b)^7(y)^7[/tex]
We have that the coefficients of the first two terms are 128 and -224.
For the first term:
=> [tex]a^7 = 128[/tex]
=> [tex]a = \sqrt[7]{128}\\ \\\\a = 2[/tex]
For the second term:
[tex]a^6b = -224[/tex]
[tex]b = \frac{-224}{a^6}[/tex]
[tex]b = \frac{-224}{2^6} \\\\\\b = \frac{-224}{64} \\\\\\b = 3.5[/tex]
Therefore, a = 2, b = 3.5
