The variable x has the fixed value 7.The variables x and y satisfy the equation \(\mathrm{4(x - 1) + 2y...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x = 7}\) (this is a fixed, known value)
  • The equation \(\mathrm{4(x - 1) + 2y = 30}\) must be satisfied
  • We need to find the value of y

2. INFER the approach

• Since we know \(\mathrm{x = 7}\), we can substitute this value directly into the equation
• This will give us an equation with only y as the unknown variable

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{x = 7}\): \(\mathrm{4(7 - 1) + 2y = 30}\)
• Work inside parentheses first: \(\mathrm{7 - 1 = 6}\)
• So we have: \(\mathrm{4(6) + 2y = 30}\)
• Multiply: \(\mathrm{24 + 2y = 30}\)
• Subtract 24 from both sides: \(\mathrm{2y = 6}\)
• Divide by 2: \(\mathrm{y = 3}\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make arithmetic errors while working through the algebraic steps

Many students correctly set up \(\mathrm{4(7 - 1) + 2y = 30}\) but then make mistakes like:

• Calculating \(\mathrm{7 - 1 = 8}\) instead of 6
• Computing \(\mathrm{4 \times 6 = 20}\) instead of 24
• Making sign errors when isolating 2y

These calculation errors lead to incorrect final answers, causing students to second-guess their approach even when their method is correct.

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students don't fully understand what 'x has the fixed value 7' means in context

Some students see two pieces of information (\(\mathrm{x = 7}\) and the equation) but don't realize they should substitute the first into the second. They might try to solve the equation \(\mathrm{4(x - 1) + 2y = 30}\) as if both x and y are unknown, leading to confusion since they have one equation with two variables.

This leads to abandoning systematic solution and guessing.

The Bottom Line:

This problem tests whether students can execute a straightforward substitution and then carefully work through several algebraic steps. The concept is simple, but the execution requires attention to arithmetic detail at each step.