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

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 \( x = -1 \).