In the xy-plane, how many points do the graphs of the lines 2x + y = 6 and 4x +...

1. TRANSLATE the problem information

• Given information:
  • Line 1: \(2\mathrm{x} + \mathrm{y} = 6\)
  • Line 2: \(4\mathrm{x} + 2\mathrm{y} = 8\)
  • Need to find: Number of intersection points

2. INFER the solution approach

• Finding intersection points means solving the system of equations
• Two main approaches: elimination method or slope-intercept comparison
• Let's try elimination first since the coefficients work nicely

3. SIMPLIFY using elimination method

• Multiply first equation by 2 to match coefficients:
  \(2(2\mathrm{x} + \mathrm{y}) = 2(6)\)
  \(4\mathrm{x} + 2\mathrm{y} = 12\)
• Now compare:
  • \(4\mathrm{x} + 2\mathrm{y} = 12\) (from first equation)
  • \(4\mathrm{x} + 2\mathrm{y} = 8\) (second equation)

4. INFER what the contradiction means

• Same left sides but different right sides: \(12 \neq 8\)
• This is impossible - no values of x and y can satisfy both equations
• Therefore: No solution exists

5. Verify with slope-intercept method

• SIMPLIFY both equations to \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\) form:
  • \(2\mathrm{x} + \mathrm{y} = 6\) → \(\mathrm{y} = -2\mathrm{x} + 6\)
  • \(4\mathrm{x} + 2\mathrm{y} = 8\) → \(\mathrm{y} = -2\mathrm{x} + 4\)
• INFER: Same slope \(-2\), different y-intercepts (6 vs 4)
• These are parallel lines that never meet

Answer: (A) Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what a contradiction means in a system of equations.

When students get \(4\mathrm{x} + 2\mathrm{y} = 12\) and \(4\mathrm{x} + 2\mathrm{y} = 8\), they might try to "solve" \(12 = 8\) or think they made an algebra mistake. They don't realize this contradiction means the system has no solution. This leads to confusion and guessing, often selecting (D) Infinitely many because they associate "no solution found" with "infinite solutions."

Second Most Common Error:

Algebraic errors during SIMPLIFY: Making mistakes when converting to slope-intercept form.

Students might incorrectly solve \(4\mathrm{x} + 2\mathrm{y} = 8\) as \(\mathrm{y} = -4\mathrm{x} + 4\) instead of \(\mathrm{y} = -2\mathrm{x} + 4\), leading them to think the lines have different slopes and intersect at exactly one point. This may lead them to select Choice (B) Exactly one.

The Bottom Line:

This problem tests whether students understand that contradictions in algebra mean "no solution exists" and can connect this abstract concept to the geometric reality that parallel lines never intersect.