\( 43 _ { x } = 61 _ { 5 } \)

To solve the equation \( 43 _ { x } = 61 _ { 5 } \), let's first convert both numbers into base 10. For the number \( 61 _ { 5 } \), we calculate: \[ 61 _ { 5 } = 6 \times 5^1 + 1 \times 5^0 = 30 + 1 = 31. \] Now, let’s express \( 43 _ { x } \) in base 10: \[ 43 _ { x } = 4 \times x^1 + 3 \times x^0 = 4x + 3. \] Setting the two base 10 values equal gives us the equation: \[ 4x + 3 = 31. \] Now, solving for \( x \): \[ 4x = 31 - 3 = 28, \] \[ x = \frac{28}{4} = 7. \] Thus, \( x = 7 \). 👓 Fun Fact: Base systems have been around for thousands of years, with the ancient Sumerians using the base 60 system, which is why we still have 60 seconds in a minute and 360 degrees in a circle today! 🚀 Real-World Application: Understanding different base systems can be crucial for computer programming and digital electronics, where binary (base 2), octal (base 8), and hexadecimal (base 16) systems are commonly used to represent data, making it easier to manage and manipulate information.