Show that $$ 4.3^{1-x}+3^{2-x}=\frac{21}{3^{x}} $$ Hence or otherwise solve for $$ x $$ if $$ 4.3^{1-x}+3^{2-x}=63 $$

Question

UpStudy AI · Accepted Answer

To solve the equation $$ 4 \cdot 3^{1-x} + 3^{2-x} = 63 $$, let's first simplify the left-hand side and show that it equals $$ \frac{21}{3^x} $$.

### Step 1: Simplify $$ 4 \cdot 3^{1-x} + 3^{2-x} $$

1. **Break down the exponents:**
   $$
   4 \cdot 3^{1-x} = 4 \cdot 3 \cdot 3^{-x} = 12 \cdot 3^{-x}
   $$
   $$
   3^{2-x} = 9 \cdot 3^{-x}
   $$

2. **Combine the terms:**
   $$
   4 \cdot 3^{1-x} + 3^{2-x} = 12 \cdot 3^{-x} + 9 \cdot 3^{-x} = (12 + 9) \cdot 3^{-x} = 21 \cdot 3^{-x}
   $$

3. **Express using positive exponents:**
   $$
   21 \cdot 3^{-x} = \frac{21}{3^x}
   $$

So, we've shown that:
$$
4 \cdot 3^{1-x} + 3^{2-x} = \frac{21}{3^x}
$$

### Step 2: Solve the Equation $$ \frac{21}{3^x} = 63 $$

1. **Set up the equation:**
   $$
   \frac{21}{3^x} = 63
   $$

2. **Multiply both sides by $$ 3^x $$:**
   $$
   21 = 63 \cdot 3^x
   $$

3. **Divide both sides by 63:**
   $$
   \frac{21}{63} = 3^x \quad \Rightarrow \quad \frac{1}{3} = 3^x
   $$

4. **Express $$ \frac{1}{3} $$ as $$ 3^{-1} $$:**
   $$
   3^{-1} = 3^x
   $$

5. **Equate the exponents:**
   $$
   x = -1
   $$

### **Final Answer**

The solution to the equation is $$ x = -1 $$.
$$ x = -1 $$

Show that \( 4.3^{1-x}+3^{2-x}=\frac{21}{3^{x}} \) Hence or otherwise solve for \( x \) if \( 4.3^{1-x}+3^{2-x}=63 \)