Solve the following equation:
$\begin{array}{l}
\frac{9}{8}(4 i+64)=\frac{5}{8} i+\frac{731}{8} \\
i=
\end{array}$
The problem presents a linear equation involving the variable \(i\):
\[\frac{9}{8}(4i + 64) = \frac{5}{8}i + \frac{731}{8}\]
Our objective is to isolate \(i\) and solve for its value. Since all terms are divided by 8, we can simplify the equation by multiplying every term on both sides by 8, which effectively clears the denominators.
Multiplying the entire equation \(\frac{9}{8}(4i + 64) = \frac{5}{8}i + \frac{731}{8}\) by 8 gives us:
\[9(4i + 64) = 5i + 731\]
Next, we distribute the 9 on the left-hand side:
\[36i + 576 = 5i + 731\]
Now, we collect the terms involving \(i\) on one side by subtracting \(5i\) from both sides:
\[36i - 5i + 576 = 731\]
\[31i + 576 = 731\]
To isolate the term with \(i\), we subtract 576 from both sides of the equation:
\[31i = 731 - 576\]
\[31i = 155\]
Finally, we divide both sides by 31 to find the value of \(i\):
\[i = \frac{155}{31} = 5\]
The value of \(i\) that satisfies the original equation is 5.
Equations like this are commonly used in finance and engineering to find an unknown interest rate or a specific physical parameter. For example, if you were calculating the break-even point for an investment where interest is paid in installments, you might set up an equation where the total accrued interest is equal to a fixed amount plus a variable portion dependent on the rate, allowing you to solve for the exact rate needed to achieve a target return. Understanding how to simplify and solve linear algebraic equations is a foundational skill for managing and optimizing these types of quantitative problems.