-{12x + 14y = 36} -{6x + 7y = -18} How many solutions does the given system of equations have?...

1. TRANSLATE the problem information

• Given system:
  • \(-12\mathrm{x} + 14\mathrm{y} = 36\)
  • \(-6\mathrm{x} + 7\mathrm{y} = -18\)
• Need to find: How many solutions this system has

2. INFER the solution approach

• To determine number of solutions, I can compare the slopes and y-intercepts of both lines
• If slopes are different → one solution (lines intersect)
• If slopes are same but y-intercepts different → zero solutions (parallel lines)
• If slopes and y-intercepts are the same → infinitely many solutions (same line)

3. SIMPLIFY both equations to slope-intercept form

First equation: \(-12\mathrm{x} + 14\mathrm{y} = 36\)

• Add 12x to both sides: \(14\mathrm{y} = 12\mathrm{x} + 36\)
• Divide everything by 14: \(\mathrm{y} = \frac{12}{14}\mathrm{x} + \frac{36}{14}\)
• Reduce fractions: \(\mathrm{y} = \frac{6}{7}\mathrm{x} + \frac{18}{7}\)

Second equation: \(-6\mathrm{x} + 7\mathrm{y} = -18\)

• Add 6x to both sides: \(7\mathrm{y} = 6\mathrm{x} - 18\)
• Divide everything by 7: \(\mathrm{y} = \frac{6}{7}\mathrm{x} - \frac{18}{7}\)

4. INFER the relationship between the lines

• Both equations have slope = \(\frac{6}{7}\) (same slope)
• Y-intercepts are \(\frac{18}{7}\) and \(-\frac{18}{7}\) (different y-intercepts)
• Same slope + different y-intercepts = parallel distinct lines = zero solutions

Answer: D (Zero solutions)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly convert to slope-intercept form but don't recognize what same slopes with different y-intercepts means.

They might think "same slope means same line" and conclude there are infinitely many solutions, forgetting to check the y-intercepts. Or they might get confused about which combination of slopes/intercepts leads to which type of solution.

This may lead them to select Choice C (Infinitely many solutions).

Second Most Common Error Path:

Poor SIMPLIFY execution: Students make algebraic errors when converting to slope-intercept form.

Common mistakes include sign errors when moving terms, or arithmetic errors when dividing by coefficients. For example, getting the wrong slope or y-intercept values, which then leads to incorrect conclusions about the relationship between the lines.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

This problem tests whether students truly understand the geometric meaning of slope and y-intercept in determining how lines relate to each other. The algebraic work is straightforward, but the conceptual leap from "same slope, different y-intercept" to "zero solutions" is where many students stumble.