Maximize \( 10b + c \) by setting \( b = 9 \), \( c = 7 \), yielding \( n = 100 imes 9 + 10 imes 9 + 7 = 997 \).
["# Maximize the Expression ( 10b + c ) Using Optimal Values of ( b ) and ( c )", "When tasked with maximizing the expression ( 10b + c ), selecting appropriate values for the variables is key to achieving the highest possible result. In this case, setting ( b = 9 ) and ( c = 7 ) delivers an optimal value:", "[\nn = 10b + c = 10 \ imes 9 + 7 = 90 + 7 = 997\n]", "## Why ( b = 9 ) and ( c = 7 ) Is Ideal", "The expression ( 10b + c ) is fundamentally structured as a two-digit number where ( b ) represents the tens digit and ( c ) the units digit. Since ( b ) contributes ten times more than ( c ), maximizing ( b ) yields the greatest numeric return. The digits ( b ) and ( c ) are constrained to values between 0 and 9 (inclusive), so the maximum allowable value for ( b ) is 9.", "By choosing ( b = 9 ), the tens part becomes ( 90 ), which is the largest single-digit contribution possible. Adding the largest single-digit value ( c = 7 ) completes the number:", "[\nn = 10 \ imes 9 + 7 = 997\n]", "### The Mathematical Breakdown", "- ( 10b ) accounts for the tens contribution\n- ( + c ) adds the units digit\n- Maximizing ( b ) first improves the total significantly due to positional weight", "Using smaller values for ( b ) or ( c )—such as ( b = 8 ) with ( c = 9 )—would yield a smaller total: ( 10 \ imes 8 + 9 = 89 ), much less than 997.", "## Practical Implications and Applications", "This maximization strategy applies in various real-world contexts:", "- Numerical optimization: When designing identification codes, prices, or secure identifiers, setting high primary digits maximizes value expressions.\n- Resource allocation: In budgeting or time management, allocating primary focus or high-impact factors first enhances overall effectiveness.\n- Coding systems: Building efficient numbering systems where weight by place matters allows optimized data structuring.", "## Conclusion: The Power of Strategic Digit Selection", "To maximize ( 10b + c ), the best choice is clearly ( b = 9 ) and ( c = 7 ), producing the maximum value of 997. Leveraging the positional strength of digits ensures optimal results in both mathematical expressions and applied systems. Remember: in numerical optimization, smart digit selection unlocks greater efficiency and value.", "---", "Keywords: maximize ( 10b + c ), ( b = 9 ), ( c = 7 ), numerical optimization, maximize expression, place value strategy, choose digits wisely, coding system optimization, maximize identifier value."]
