\(3\mathrm{x} + 5(\mathrm{x} + 4) = 76\) What value of x is the solution to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{3x + 5(x + 4) = 76}\)
• Need to find: value of \(\mathrm{x}\)

2. INFER the solution strategy

• The equation has parentheses, so distribute first
• Then combine like terms and isolate \(\mathrm{x}\) using inverse operations
• This follows the standard order for solving linear equations

3. SIMPLIFY by applying the distributive property

• \(\mathrm{5(x + 4) = 5x + 20}\)
• Equation becomes: \(\mathrm{3x + 5x + 20 = 76}\)

4. SIMPLIFY by combining like terms

• \(\mathrm{3x + 5x = 8x}\)
• Equation becomes: \(\mathrm{8x + 20 = 76}\)

5. SIMPLIFY by isolating the variable term

• Subtract 20 from both sides: \(\mathrm{8x = 56}\)

6. SIMPLIFY to find the final answer

• Divide both sides by 8: \(\mathrm{x = 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors during the distribution step or when combining like terms.

For example, they might incorrectly distribute as \(\mathrm{5(x + 4) = 5x + 4}\) (forgetting to multiply \(\mathrm{5 \times 4}\)), leading to \(\mathrm{3x + 5x + 4 = 76}\), then \(\mathrm{8x = 72}\), and finally \(\mathrm{x = 9}\). Since 9 isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly work through the problem until they get \(\mathrm{8x = 56}\), but then stop without completing the final division step.

They see that 56 appears as Choice C and select it, not realizing they need to divide by 8 to get the actual value of \(\mathrm{x}\). This may lead them to select Choice C (56).

Third Most Common Error:

Poor INFER reasoning about equation setup: Students misinterpret the original equation structure and solve a simpler version instead.

As noted in the solution explanation, some students might solve \(\mathrm{x + 4 = 76}\) instead of the full equation, getting \(\mathrm{x = 72}\). This may lead them to select Choice D (72).

The Bottom Line:

This problem tests whether students can systematically work through multiple algebraic steps without losing focus on what the variable actually represents. The key is completing every step of the simplification process, not stopping when you see a familiar number in the answer choices.