\(5(\mathrm{t} + 3) - 7(\mathrm{t} + 3) = 38\) What value of t is the solution to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(5(\mathrm{t} + 3) - 7(\mathrm{t} + 3) = 38\)
• Need to find: The value of t

2. INFER the most efficient approach

• Notice that \((\mathrm{t} + 3)\) appears as a factor in both terms on the left side
• Instead of distributing first, we can factor out the common term \((\mathrm{t} + 3)\)
• This will create a simpler equation to solve

3. SIMPLIFY by factoring out the common term

• Factor out \((\mathrm{t} + 3)\): \((\mathrm{t} + 3)[5 - 7] = 38\)
• Combine the coefficients: \((\mathrm{t} + 3)(-2) = 38\)
• Rewrite: \(-2(\mathrm{t} + 3) = 38\)

4. SIMPLIFY to solve for the expression in parentheses

• Divide both sides by -2: \(\mathrm{t} + 3 = -19\)
• Be careful with the sign: \(38 \div (-2) = -19\)

5. SIMPLIFY to find the final value

• Subtract 3 from both sides: \(\mathrm{t} = -22\)

Answer: -22 or t = -22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the factoring opportunity and instead distribute each term first, creating: \(5\mathrm{t} + 15 - 7\mathrm{t} - 21 = 38\). While this approach works, it involves more steps and creates more opportunities for arithmetic errors, especially when combining like terms and working with multiple negative numbers.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when dividing by -2 or when working with the negative coefficients. For example, they might get \(\mathrm{t} + 3 = 19\) instead of \(\mathrm{t} + 3 = -19\), leading to \(\mathrm{t} = 16\) as their final answer.

The Bottom Line:

This problem rewards students who can spot patterns and choose efficient solution paths. The key insight is recognizing that factoring first makes the algebra much cleaner than distributing first.