The ratio of x to y is -{3} to 4. Also, 2x + 5y = 14. What is the solution...

1. TRANSLATE the ratio information

• Given information:
  • Ratio of x to y is -3 to 4
  • \(\mathrm{2x + 5y = 14}\)

• The ratio \(\mathrm{x:y = -3:4}\) means \(\mathrm{x/y = -3/4}\), so \(\mathrm{x = -3y/4}\)

2. INFER the solution strategy

• We have one equation already expressing x in terms of y
• We can substitute this expression into the linear equation
• This will give us one equation with one variable (y)

3. SIMPLIFY through substitution

• Substitute \(\mathrm{x = -3y/4}\) into \(\mathrm{2x + 5y = 14}\):

\(\mathrm{2(-3y/4) + 5y = 14}\)
\(\mathrm{-6y/4 + 5y = 14}\)
\(\mathrm{-3y/2 + 5y = 14}\)

4. SIMPLIFY the fraction arithmetic

• Convert 5y to halves: \(\mathrm{5y = 10y/2}\)
• Combine: \(\mathrm{-3y/2 + 10y/2 = 7y/2}\)
• So: \(\mathrm{7y/2 = 14}\)
• Multiply both sides by 2: \(\mathrm{7y = 28}\)
• Therefore: \(\mathrm{y = 4}\)

5. Find x using the ratio relationship

\(\mathrm{x = -3y/4 = -3(4)/4 = -3}\)

Answer: (-3, 4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often misinterpret the ratio "x to y is -3 to 4" and write \(\mathrm{x = -3, y = 4}\) directly, thinking this IS the solution rather than a relationship.

They then check: \(\mathrm{2(-3) + 5(4) = -6 + 20 = 14}\) ✓, and since this works, they conclude the answer is (-3, 4) without realizing they got lucky. However, if the second equation had different constants, this approach would fail completely.

This may lead them to select Choice B ((-3, 4)) for the wrong reasons, getting the right answer through flawed logic.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x = -3y/4}\) and substitute, but make fraction arithmetic errors when combining \(\mathrm{-3y/2 + 5y}\). They might forget to convert 5y to the common denominator of halves, getting expressions like \(\mathrm{-3y/2 + 5y = 2y/2}\) instead of \(\mathrm{7y/2}\).

This leads to wrong values for y, and subsequently wrong values for x, causing them to select Choice A ((-6, 8)) or Choice D ((6, -8)).

The Bottom Line:

This problem tests whether students can properly translate ratio language into algebraic relationships and then execute substitution method accurately. The key insight is recognizing that ratios give us relationships between variables, not the actual values of the variables.