Let's solve the given system of equations step-by-step.
The system of equations is:
[tex]\[ 2^{3x} + y \times 3^{4x^{-y}} = 8480 \tag{1} \][/tex]
[tex]\[ 2^{3x + y} \times 3^{4x - y} = 648 \tag{2} \][/tex]
First, we'll look at the second equation to identify some key observations and possible simplifications.
### Step 1: Analyzing the Second Equation
[tex]\[ 2^{3x + y} \times 3^{4x - y} = 648 \][/tex]
Notice that 648 can be factored into primes:
[tex]\[ 648 = 2^3 \times 3^4 \][/tex]
We now rewrite the second equation as:
[tex]\[ 2^{3x + y} \times 3^{4x - y} = 2^3 \times 3^4 \][/tex]
### Step 2: Equating Exponents
Since the bases are powers of 2 and 3, we can equate the exponents:
[tex]\[ 3x + y = 3 \tag{3} \][/tex]
[tex]\[ 4x - y = 4 \tag{4} \][/tex]
We now have a linear system in equations (3) and (4).
### Step 3: Solving the Linear System
We first solve equation (3) for [tex]\( y \)[/tex]:
[tex]\[ y = 3 - 3x \tag{5} \][/tex]
Now substitute [tex]\( y \)[/tex] from equation (5) into equation (4):
[tex]\[ 4x - (3 - 3x) = 4 \][/tex]
[tex]\[ 4x - 3 + 3x = 4 \][/tex]
[tex]\[ 7x - 3 = 4 \][/tex]
[tex]\[ 7x = 7 \][/tex]
[tex]\[ x = 1 \][/tex]
Once we have [tex]\( x = 1 \)[/tex], we substitute back into equation (5) to find [tex]\( y \)[/tex]:
[tex]\[ y = 3 - 3(1) \][/tex]
[tex]\[ y = 0 \][/tex]
### Step 4: Verify the Solutions
We substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 0 \)[/tex] back into the original equations to verify.
For the first equation:
[tex]\[ 2^{3(1)} + 0 \times 3^{4(1)^{0}} = 8480 \][/tex]
[tex]\[ 2^3 + 0 \times 3^4 = 8 \neq 8480 \][/tex]
Clearly, this doesn't satisfy the first equation. Thus, there is an inconsistency, and we need to check these steps. Let's go back and revisit if there could be any missed steps or implicit assumptions to correct from here.
### Reflection on Correcting Approach
In our problem, inherently after solving the system for values it mismatches initial variables. Reevaluating system aligns simple constraints but indicates additional factors or nonlinear dependencies might need addressing. Initially our linear assumptions solved but iteration steps factor rechecking, solving by corrections can unveil complexities more accurately without departure, validating or reassessment loop sought after:
If initial mismatch errors are right might deeper symbolic tools aiding contribute evaluation further or reassert solutions polynomially precise deeper structurally examining outcome.
