Find the values of base \( x \) of the follow i. \( 3(14 x)=45 x \) ii. \( 314 x=103_{a} \)

To solve the equations for the base \( x \), we need to look at each equation separately. For the first equation \( 3(14_x) = 45_x \): 1. Convert both sides from base \( x \) to decimal: - \( 14_x = 1 \cdot x + 4 \cdot 1 = x + 4 \) - \( 45_x = 4 \cdot x + 5 \cdot 1 = 4x + 5 \) 2. Substitute back into the equation: \[ 3(x + 4) = 4x + 5 \] Simplifying gives: \[ 3x + 12 = 4x + 5 \] Rearranging leads to: \[ 12 - 5 = 4x - 3x \quad \Rightarrow \quad 7 = x \] Thus, the value of base \( x \) for the first equation is \( 7 \). For the second equation \( 314_x = 103_a \): 1. Convert \( 314_x \) from base \( x \) to decimal: - \( 314_x = 3 \cdot x^2 + 1 \cdot x + 4 = 3x^2 + x + 4 \) 2. Convert \( 103_a \) from base \( a \) to decimal: - \( 103_a = 1 \cdot a^2 + 0 \cdot a + 3 = a^2 + 3 \) 3. Set the two equations equal: \[ 3x^2 + x + 4 = a^2 + 3 \] Normally, we need specific values for \( a \) or rules tying \( x \) and \( a \) to find simultaneous solutions. However, if we assume \( a \) is also determined by \( x \) as a function of the base used, we can analyze specific values for clarity, but the answer depends on the context of \( a \) in bases or constraints not explicitly stated. For example, if \( a = x \), the next steps would involve solving for \( x \) through substitution or equality checking. Please let me know if you have specific constraints or numerical values for \( a \) to proceed further!