If the set of equations $x+2y+3z=6$,$x+3y+5z=9$,and $2x+5y+az=b$ has a unique solution,then:

Examine the consistency of the system of equations: $x+2y=2$ and $2x+3y=3$.

If $(\alpha, \beta, \gamma)$ is the solution of the system of simultaneous linear equations given by $3x + 4y - 5z = -6$,$2x + 3y - 4z = -7$,and $4x - 2y + z = 9$,then find the value of $\alpha + 3\beta - 2\gamma$.

For a matrix $A=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$,if $U_{1}, U_{2}$,and $U_{3}$ are $3 \times 1$ column matrices satisfying $A U_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$,$A U_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$,$A U_{3}=\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$,and $U$ is a $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$,and $U_{3}$,then the sum of the elements of $U^{-1}$ is:

Let $a$ be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b = \lim_{x \rightarrow 0} \left( \frac{\int_0^x \frac{\ln(1+t)}{t^{2024}+1} dt}{x^2} \right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root,where $c, d, e \in \mathbb{R}$,then $d:c:e$ equals

The number of solutions of the following equations $x_2 - x_3 = 1$,$-x_1 + 2x_3 = -2$,$x_1 - 2x_2 = 3$ is