Which of the following systems of linear equations has no solution?3x - 2y = 86x - 4y = 16y =...

1. INFER the key insight about no-solution systems

A system of linear equations has no solution when the equations represent parallel but distinct lines. This happens when:

• The coefficients of x and y are proportional between equations
• But the constant terms are NOT proportional

2. SIMPLIFY each system to check for this condition

Let's examine each option systematically:

Option A: \(\mathrm{3x - 2y = 8}\) and \(\mathrm{6x - 4y = 16}\)

• Multiply the first equation by 2: \(\mathrm{6x - 4y = 16}\)
• This matches the second equation exactly
• INFER: These are the same line → infinitely many solutions

Option B: \(\mathrm{y = 7}\) and \(\mathrm{2y - 14 = x}\)

• INFER: Substitute \(\mathrm{y = 7}\) into the second equation
• \(\mathrm{2(7) - 14 = x}\)
• \(\mathrm{14 - 14 = 0 = x}\)
• One solution exists: \(\mathrm{(0, 7)}\)

Option C: \(\mathrm{5x + 10y = 15}\) and \(\mathrm{x + 2y = 4}\)

• SIMPLIFY: Multiply the second equation by 5
• \(\mathrm{5(x + 2y) = 5(4)}\)
• \(\mathrm{5x + 10y = 20}\)
• INFER: Compare with first equation: \(\mathrm{5x + 10y = 15}\)
• Same coefficients \(\mathrm{(5, 10)}\) but different constants \(\mathrm{(15 \neq 20)}\)
• This means parallel distinct lines → no solution

Option D: \(\mathrm{x = -3}\) and \(\mathrm{y = -2x - 7}\)

• INFER: Substitute \(\mathrm{x = -3}\) into the second equation
• \(\mathrm{y = -2(-3) - 7 = 6 - 7 = -1}\)
• One solution exists: \(\mathrm{(-3, -1)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse 'identical equations' with 'no solution'

Many students see that Option A has equations where one is a multiple of the other and incorrectly think this means 'no solution.' They don't recognize that identical equations actually represent the same line, giving infinitely many solutions rather than no solution.

This may lead them to select Choice A (\(\mathrm{3x - 2y = 8, 6x - 4y = 16}\))

Second Most Common Error:

Inadequate SIMPLIFY execution: Students don't manipulate equations to make comparisons clear

Students may look at Option C and not think to multiply the second equation by 5 to make the comparison obvious. Without this algebraic manipulation, they can't easily see that the coefficients match but the constants don't.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand the geometric meaning behind algebraic relationships in systems of equations. The key insight is distinguishing between equations that represent the same line versus equations that represent parallel but distinct lines.