Final answer:
To solve the simultaneous equation using Cramer's rule and inverse matrix, we rewrite the equations in standard form and represent them in matrix form. We then apply Cramer's rule to find the solutions by calculating the determinants of the matrices involved. The solutions to the simultaneous equation are $x = \frac{2}{9} \cdot 9^x$ and $y = \frac{2}{9} \cdot 9^x$
Step-by-step explanation:
To solve the simultaneous equation using Cramer's rule and inverse matrix, let's rewrite the equations in standard form:
1) $(27^x) - y = \frac{9^x}{3}$
2) $9^x - 3y = 0$
Now, let's represent these equations in matrix form:
$A\textbf{x} = \textbf{b}$
where $A = \begin{bmatrix} 27 & -1 \\ 9 & -3 \end{bmatrix}$, $\textbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}$, and $\textbf{b} = \begin{bmatrix} \frac{9^x}{3} \\ 0 \end{bmatrix}$
To find the solutions, we can apply Cramer's rule:
$x = \frac{det(A_1)}{det(A)}$ and $y = \frac{det(A_2)}{det(A)}$
where $det(A)$ is the determinant of matrix $A$, $A_1$ is obtained by replacing the first column of $A$ with $\textbf{b}$, and $A_2$ is obtained by replacing the second column of $A$ with $\textbf{b}$.
After calculating the determinants, we find that $det(A) = 36$, $det(A_1) = 9 \cdot 9^x - 3 \cdot \frac{9^x}{3} = 9^x(9-1) = 8 \cdot 9^x$, and $det(A_2) = 27 \cdot \frac{9^x}{3} - (-1) \cdot 9^x = 8 \cdot 9^x$
Therefore, $x = \frac{det(A_1)}{det(A)} = \frac{8 \cdot 9^x}{36} = \frac{2 \cdot 9^x}{9} = \frac{2}{9} \cdot 9^x$
And $y = \frac{det(A_2)}{det(A)} = \frac{8 \cdot 9^x}{36} = \frac{2 \cdot 9^x}{9} = \frac{2}{9} \cdot 9^x$
The solutions to the simultaneous equation are $x = \frac{2}{9} \cdot 9^x$ and $y = \frac{2}{9} \cdot 9^x$