5x - 2y = 8 3x + y = 7 If \((\mathrm{x}, \mathrm{y})\) is the solution to the system of...

1. TRANSLATE the problem information

• Given system:
  • \(5\mathrm{x} - 2\mathrm{y} = 8\)
  • \(3\mathrm{x} + \mathrm{y} = 7\)
• Find: \(2\mathrm{x} + 2\mathrm{y}\) (not just x and y individually)

2. INFER the solution approach

• The second equation is simpler since y has coefficient 1
• Use substitution: solve the second equation for y, then substitute into the first
• This will give us one equation in one variable (x)

3. SIMPLIFY to find y in terms of x

From \(3\mathrm{x} + \mathrm{y} = 7\):

\(\mathrm{y} = 7 - 3\mathrm{x}\)

4. SIMPLIFY by substitution to solve for x

Substitute \(\mathrm{y} = 7 - 3\mathrm{x}\) into the first equation:

\(5\mathrm{x} - 2(7 - 3\mathrm{x}) = 8\)

Distribute the -2:

\(5\mathrm{x} - 14 + 6\mathrm{x} = 8\)

Combine like terms:

\(11\mathrm{x} - 14 = 8\)

\(11\mathrm{x} = 22\)

\(\mathrm{x} = 2\)

5. SIMPLIFY to find y

Substitute \(\mathrm{x} = 2\) back into \(\mathrm{y} = 7 - 3\mathrm{x}\):

\(\mathrm{y} = 7 - 3(2) = 7 - 6 = 1\)

6. SIMPLIFY to calculate the target expression

\(2\mathrm{x} + 2\mathrm{y} = 2(2) + 2(1) = 4 + 2 = 6\)

Answer: C (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the substitution process, particularly when distributing \(-2(7 - 3\mathrm{x}) = -14 + 6\mathrm{x}\). They might get \(-2(7 - 3\mathrm{x}) = -14 - 6\mathrm{x}\) instead, leading to \(5\mathrm{x} - 14 - 6\mathrm{x} = 8\), which gives \(-\mathrm{x} - 14 = 8\), so \(\mathrm{x} = -22\).

With this wrong x-value, they get \(\mathrm{y} = 7 - 3(-22) = 73\), leading to \(2\mathrm{x} + 2\mathrm{y} = 2(-22) + 2(73) = -44 + 146 = 102\). Since this doesn't match any answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students solve correctly for \(\mathrm{x} = 2\) and \(\mathrm{y} = 1\), but then forget what the question is actually asking for. They might look for answer choices that equal \(\mathrm{x} = 2\) or \(\mathrm{y} = 1\), or even \(\mathrm{x} + \mathrm{y} = 3\), rather than calculating \(2\mathrm{x} + 2\mathrm{y} = 6\).

This causes them to get stuck since none of the choices equal 2, 1, or 3, leading to random answer selection.

The Bottom Line:

This problem tests both systematic algebraic manipulation and careful attention to what's being asked. Success requires methodical substitution work and remembering to calculate the specific expression requested, not just the individual variables.