Question

If \((2m+n) + (2n+m)=27\), find the maximum value of \((2m-3)\), assuming m and n are positive integers. 
 

Hint:

If an algebra problem seems to have no solution or an unbounded solution (no maximum/minimum), double-check for potential typos. A product might be a sum, or there might be an unstated but implied constraint, such as the variables being positive integers.

Answer 1

Correct Answer: 13

Step 1: Key Formula or Approach:
1. Simplify the corrected linear equation to establish a direct relationship between 'm' and 'n'.
2. To maximize the expression \(2m-3\), we must maximize the value of 'm'.
3. Using the relationship from step 1, maximizing 'm' will require minimizing 'n'.
4. We apply the common implicit constraint in such problems that 'm' and 'n' are positive integers.
Step 2: Detailed Explanation:
We start with the corrected equation: \[ (2m+n) + (2n+m) = 27 \] Combine like terms: \[ 3m + 3n = 27 \] Divide the entire equation by 3: \[ m + n = 9 \] Our goal is to maximize the expression \(2m-3\). This is equivalent to maximizing 'm'.
From the equation \(m + n = 9\), we can express 'm' in terms of 'n': \[ m = 9 - n \] To maximize 'm', we must choose the smallest possible value for 'n'.
Since we are assuming 'm' and 'n' are positive integers, the smallest possible value for 'n' is 1.
Let's substitute \(n = 1\) to find the maximum value of 'm': \[ m_{max} = 9 - 1 = 8 \] Now that we have the maximum value for 'm', we can find the maximum value of the target expression: \[ \text{Maximum value of } (2m-3) = 2(m_{max}) - 3 \] \[ = 2(8) - 3 \] \[ = 16 - 3 \] \[ = 13 \] Step 3: Final Answer:
Based on the logical correction of the problem statement, the maximum value of the expression is 13.