Question:The variables x and y satisfy the system of equations:2x - 3y = 1/24x + y = 13/2What is the...

1. TRANSLATE the problem information

• Given system:
  • \(2\mathrm{x} - 3\mathrm{y} = \frac{1}{2}\)
  • \(4\mathrm{x} + \mathrm{y} = \frac{13}{2}\)
• Need to find: the value of y

2. INFER the solution approach

• Since the second equation has y with coefficient 1, it's easier to solve for y in terms of x
• Use substitution method: solve the simpler equation for one variable, then substitute

3. SIMPLIFY to isolate y from the second equation

From \(4\mathrm{x} + \mathrm{y} = \frac{13}{2}\), we get:

\(\mathrm{y} = \frac{13}{2} - 4\mathrm{x}\)

4. SIMPLIFY by substituting into the first equation

Substitute \(\mathrm{y} = \frac{13}{2} - 4\mathrm{x}\) into \(2\mathrm{x} - 3\mathrm{y} = \frac{1}{2}\):

\(2\mathrm{x} - 3(\frac{13}{2} - 4\mathrm{x}) = \frac{1}{2}\)

5. SIMPLIFY using the distributive property

\(2\mathrm{x} - 3(\frac{13}{2}) - 3(-4\mathrm{x}) = \frac{1}{2}\)

\(2\mathrm{x} - \frac{39}{2} + 12\mathrm{x} = \frac{1}{2}\)

\(14\mathrm{x} - \frac{39}{2} = \frac{1}{2}\)

6. SIMPLIFY to solve for x

\(14\mathrm{x} = \frac{1}{2} + \frac{39}{2} = \frac{40}{2} = 20\)

\(\mathrm{x} = \frac{20}{14} = \frac{10}{7}\)

7. SIMPLIFY to find y by substitution

\(\mathrm{y} = \frac{13}{2} - 4(\frac{10}{7})\)

\(\mathrm{y} = \frac{13}{2} - \frac{40}{7}\)

Convert to common denominator:

\(\mathrm{y} = \frac{91}{14} - \frac{80}{14} = \frac{11}{14}\)

Answer: \(\frac{11}{14}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when distributing the negative coefficient

When expanding \(2\mathrm{x} - 3(\frac{13}{2} - 4\mathrm{x})\), students often make the error:

\(2\mathrm{x} - \frac{39}{2} - 12\mathrm{x} = \frac{1}{2}\) (forgetting that \(-3 \times (-4\mathrm{x}) = +12\mathrm{x}\))

This leads to: \(-10\mathrm{x} = \frac{1}{2} + \frac{39}{2} = 20\), so \(\mathrm{x} = -2\)

Then \(\mathrm{y} = \frac{13}{2} - 4(-2) = \frac{13}{2} + 8 = \frac{29}{2}\)

This type of error leads to confusion as verification fails, causing students to get stuck and guess.

Second Most Common Error:

Weak SIMPLIFY skill: Fraction arithmetic errors when finding common denominators

Students might correctly get to \(\mathrm{y} = \frac{13}{2} - \frac{40}{7}\) but then make errors like:

• Using wrong common denominator (7 instead of 14)
• Arithmetic errors: \(\frac{91}{14} - \frac{80}{14} = \frac{71}{14}\) instead of \(\frac{11}{14}\)

These calculation errors produce incorrect final answers that don't verify with the original system.

The Bottom Line:

This problem tests careful algebraic manipulation with fractions. Success requires methodical application of the distributive property and precise fraction arithmetic—areas where small errors compound into completely wrong answers.