2x - 4y = 8 x - 2y = 7 At how many points do the graphs of the given...

1. TRANSLATE the problem information

• Given information:
  • Two linear equations: \(\mathrm{2x - 4y = 8}\) and \(\mathrm{x - 2y = 7}\)
  • Need to find number of intersection points

2. SIMPLIFY the first equation

• Divide all terms in \(\mathrm{2x - 4y = 8}\) by 2:

\(\mathrm{2x \div 2 - 4y \div 2 = 8 \div 2}\)

\(\mathrm{x - 2y = 4}\)

3. INFER the relationship between equations

• Now we have the system:
  • \(\mathrm{x - 2y = 4}\)
  • \(\mathrm{x - 2y = 7}\)
• The same expression \(\mathrm{(x - 2y)}\) cannot equal two different values simultaneously
• This means the equations represent parallel lines

4. INFER the geometric meaning

• Parallel lines have identical slopes but different y-intercepts
• Parallel lines never intersect
• Therefore: zero intersection points

Answer: (A) Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students may attempt to solve the original system algebraically without first simplifying, leading to unnecessarily complex calculations that obscure the key insight.

They might try elimination or substitution on the original equations, get confused by the arithmetic, and either make calculation errors or fail to recognize that their work is leading to a contradiction like \(\mathrm{0 = 3}\) (which would indicate no solution).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning about contradictions: Even students who simplify correctly may not understand what it means when they get \(\mathrm{x - 2y = 4}\) and \(\mathrm{x - 2y = 7}\).

They might think this means there are two different solutions rather than recognizing it as an impossible situation. They may attempt to continue solving algebraically instead of stopping to consider the geometric interpretation.

This may lead them to select Choice (D) (Infinitely many), thinking that having "two equations" somehow creates infinite solutions.

The Bottom Line:

The key insight is recognizing that when the same algebraic expression equals two different constants, you have parallel lines that never meet. Students who focus only on mechanical algebra without considering the geometric meaning often miss this crucial point.