3/5x + 3/4y = 7 Which table gives three values of x and their corresponding values of y for the...

1. TRANSLATE the problem requirements

• Given information:
  • Linear equation: \(\frac{3}{5}x + \frac{3}{4}y = 7\)
  • Need to verify which table correctly shows x-values and corresponding y-values
• Strategy: Substitute each x-value (1, 2, 4) into the equation and solve for y

2. SIMPLIFY for x = 1

• Substitute: \(\frac{3}{5}(1) + \frac{3}{4}y = 7\)
• Simplify: \(\frac{3}{5} + \frac{3}{4}y = 7\)
• Isolate the y-term: \(\frac{3}{4}y = 7 - \frac{3}{5}\)
• Convert to common denominator: \(\frac{3}{4}y = \frac{35}{5} - \frac{3}{5} = \frac{32}{5}\)
• Solve for y: \(y = \frac{32}{5} \times \frac{4}{3} = \frac{128}{15}\)

3. SIMPLIFY for x = 2

• Substitute: \(\frac{3}{5}(2) + \frac{3}{4}y = 7\)
• Simplify: \(\frac{6}{5} + \frac{3}{4}y = 7\)
• Isolate the y-term: \(\frac{3}{4}y = 7 - \frac{6}{5}\)
• Convert to common denominator: \(\frac{3}{4}y = \frac{35}{5} - \frac{6}{5} = \frac{29}{5}\)
• Solve for y: \(y = \frac{29}{5} \times \frac{4}{3} = \frac{116}{15}\)

4. SIMPLIFY for x = 4

• Substitute: \(\frac{3}{5}(4) + \frac{3}{4}y = 7\)
• Simplify: \(\frac{12}{5} + \frac{3}{4}y = 7\)
• Isolate the y-term: \(\frac{3}{4}y = 7 - \frac{12}{5}\)
• Convert to common denominator: \(\frac{3}{4}y = \frac{35}{5} - \frac{12}{5} = \frac{23}{5}\)
• Solve for y: \(y = \frac{23}{5} \times \frac{4}{3} = \frac{92}{15}\)

5. Match results to answer choices

The correct table should show:

• x = 1, y = 128/15
• x = 2, y = 116/15
• x = 4, y = 92/15

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when working with fractions, particularly when finding common denominators or multiplying by reciprocals.

For example, when solving \(\frac{3}{4}y = \frac{32}{5}\), they might incorrectly multiply as \(y = \frac{32}{5} \times \frac{3}{4}\) instead of \(y = \frac{32}{5} \times \frac{4}{3}\), getting \(y = \frac{96}{20} = \frac{24}{5}\) instead of \(\frac{128}{15}\). This systematic error across all three calculations would lead them to select a consistently incorrect table, or they might get confused by their inconsistent results and guess.

Second Most Common Error:

Poor fraction arithmetic: Students struggle with converting 7 to fifths when subtracting fractions, writing \(7 - \frac{3}{5} = \frac{4}{5}\) instead of correctly getting \(\frac{35}{5} - \frac{3}{5} = \frac{32}{5}\).

This leads to incorrect y-values that don't match any of the given tables exactly. This may lead them to select the table with values closest to their incorrect calculations, or abandon the systematic approach and guess.

The Bottom Line:

This problem tests systematic algebraic thinking combined with careful fraction arithmetic. Success requires maintaining accuracy through repetitive calculations while managing multiple fraction operations in each step.