Certainly! Let's solve the given equation step-by-step:
The equation we are given is:
[tex]\[ 7 + 42 \cdot 3^{2 - 3a} = 14 \cdot 3^{-2a} + 7 \][/tex]
### Step 1: Simplify Both Sides of the Equation
First, let's eliminate the 7 terms on both sides:
[tex]\[ 42 \cdot 3^{2 - 3a} = 14 \cdot 3^{-2a} \][/tex]
### Step 2: Divide Both Sides by the Common Factor
Next, let's divide both sides of the equation by 14 to make it simpler:
[tex]\[ 3 \cdot 3^{2 - 3a} = 3^{-2a} \][/tex]
### Step 3: Use Properties of Exponents
We can rewrite [tex]\( 3 \cdot 3^{2 - 3a} \)[/tex] as:
[tex]\[ 3^1 \cdot 3^{2 - 3a} = 3^{1 + 2 - 3a} = 3^{3 - 3a} \][/tex]
So, the equation now becomes:
[tex]\[ 3^{3 - 3a} = 3^{-2a} \][/tex]
### Step 4: Equate the Exponents
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 3 - 3a = -2a \][/tex]
### Step 5: Solve for [tex]\(a\)[/tex]
Now we solve the equation for [tex]\(a\)[/tex]:
[tex]\[ 3 - 3a = -2a \][/tex]
Add [tex]\(3a\)[/tex] to both sides:
[tex]\[ 3 = a \][/tex]
So, we have found that:
[tex]\[ a = 3 \][/tex]
### Conclusion
Among the given possible solutions [tex]\(a = -3\)[/tex], [tex]\(a = 0\)[/tex], [tex]\(a = 3\)[/tex], and "no solution", the solution to the equation is:
[tex]\[ a = 3 \][/tex]
Therefore, the valid answer is:
[tex]\[ a = 3 \][/tex]
