Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1$. If $u$ and $v$ satisfy the equations $\begin{aligned} \alpha u+\beta v &= 18 \\ \gamma u+\delta v &= 20 \end{aligned}$ then $u+v$ equals:

Step 1: Simplify the given expression

We use the binomial expansion formula for $(A+B)^n + (A-B)^n$. For $n=5$, the terms with odd powers of $B$ cancel out. Here, $A=x$ and $B=\sqrt{x^3-1}$.

Step 2: Substitute A and B into the simplified expression

Substitute $A=x$ and $B=\sqrt{x^3-1}$ into the simplified expression. Then, expand and combine like terms to get the polynomial in descending powers of $x$.

Step 3: Identify the coefficients

From the expanded polynomial, we identify the coefficients of $x^7, x^5, x^3,$ and $x$. The coefficient of $x^7$ is $\alpha$, $x^5$ is $\beta$, $x^3$ is $\gamma$, and $x$ is $\delta$.

Step 4: Substitute coefficients into the given equations

Substitute the values of $\alpha, \beta, \gamma,$ and $\delta$ into the given system of linear equations.

Step 5: Solve the system of equations for u and v

Divide equation (1) by 2 to simplify it to $5u+v=9$. Divide equation (2) by 10 to simplify it to $-2u+v=2$. Subtract the second simplified equation from the first to find $u$. Then substitute the value of $u$ back into one of the simplified equations to find $v$.

Step 6: Calculate u+v

Finally, add the calculated values of $u$ and $v$ to find their sum.

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