3x = 36y - 45One of the two equations in a system of linear equations is given. The system has...

1. TRANSLATE the problem requirements

• Given: One equation \(\mathrm{3x = 36y - 45}\) in a system
• Need: Second equation that creates no solution
• What this means: We need parallel but distinct lines (same slope, different y-intercepts)

2. TRANSLATE the given equation to slope-intercept form

• Start with: \(\mathrm{3x = 36y - 45}\)
• Add 45 to both sides: \(\mathrm{3x + 45 = 36y}\)
• Divide by 36: \(\mathrm{y = \frac{3x + 45}{36}}\)
• SIMPLIFY: \(\mathrm{y = \frac{1}{12}x + \frac{5}{4}}\)
• The given line has \(\mathrm{slope = \frac{1}{12}}\) and \(\mathrm{y\text{-}intercept = \frac{5}{4}}\)

3. INFER what we need in the second equation

• For no solution: Need same slope \(\mathrm{\left(\frac{1}{12}\right)}\) but different y-intercept
• For one solution: Need different slope
• For infinite solutions: Need identical slope and y-intercept

4. TRANSLATE and SIMPLIFY each answer choice

Choice A: \(\mathrm{x = 4y}\) → \(\mathrm{y = \frac{1}{4}x}\)

• \(\mathrm{Slope = \frac{1}{4}}\), \(\mathrm{y\text{-}intercept = 0}\)

Choice B: \(\mathrm{\frac{1}{3}x = 4y}\) → \(\mathrm{y = \frac{1}{12}x}\)

• \(\mathrm{Slope = \frac{1}{12}}\), \(\mathrm{y\text{-}intercept = 0}\)

Choice C: \(\mathrm{x = 12y - 15}\) → \(\mathrm{12y = x + 15}\) → \(\mathrm{y = \frac{1}{12}x + \frac{5}{4}}\)

• \(\mathrm{Slope = \frac{1}{12}}\), \(\mathrm{y\text{-}intercept = \frac{5}{4}}\)

Choice D: \(\mathrm{\frac{1}{3}x = 12y - 15}\) → \(\mathrm{12y = \frac{1}{3}x + 15}\) → \(\mathrm{y = \frac{1}{36}x + \frac{5}{4}}\)

• \(\mathrm{Slope = \frac{1}{36}}\), \(\mathrm{y\text{-}intercept = \frac{5}{4}}\)

5. INFER which choice gives no solution

• Choice B has \(\mathrm{slope = \frac{1}{12}}\) (same as given) and \(\mathrm{y\text{-}intercept = 0}\) (different from \(\mathrm{\frac{5}{4}}\))
• Same slope + different y-intercept = parallel but distinct lines = no solution

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students know that parallel lines have no solution, but they don't make the connection that parallel means "same slope" and distinct means "different y-intercept." They might look for equations that "look different" from the original rather than systematically comparing slopes and y-intercepts.

This leads to confusion and guessing between the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when converting equations to slope-intercept form, particularly with fractions. For example, they might incorrectly simplify \(\mathrm{\frac{1}{3}x = 4y}\) as \(\mathrm{y = \frac{1}{3}x \div 4 = \frac{1}{3}x}\) instead of \(\mathrm{y = \frac{1}{12}x}\).

This may lead them to select Choice A (\(\mathrm{\frac{1}{4}x}\)) or get confused about which slopes actually match.

The Bottom Line:

This problem tests whether students can systematically analyze linear systems by converting to standard form and comparing key features (slope and y-intercept), rather than relying on visual inspection or guesswork.