x^2 + y^2 = 9x^2 + y^2 = 25At how many points do the graphs of the given equations intersect...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{x^2 + y^2 = 9}\)
  • Second equation: \(\mathrm{x^2 + y^2 = 25}\)
• What this tells us: Both equations represent circles centered at the origin

2. INFER the approach

• To find intersection points, we need values of x and y that satisfy BOTH equations simultaneously
• We can approach this algebraically by manipulating the equations
• Key insight: If both equations are true at the same point, we can set them equal or subtract one from the other

3. SIMPLIFY using algebraic manipulation

• Subtract the first equation from the second:
  \(\mathrm{(x^2 + y^2) - (x^2 + y^2) = 25 - 9}\)
• This simplifies to: \(\mathrm{0 = 16}\)

4. INFER the meaning of the result

• The equation \(\mathrm{0 = 16}\) is a contradiction - it's never true
• Since this contradiction arose from assuming the equations have common solutions, there can be no intersection points
• Geometric interpretation: These are concentric circles with different radii (3 and 5), so they cannot intersect

Answer: A (Zero)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{0 = 16}\) is a contradiction that means "no solutions"

Students might see \(\mathrm{0 = 16}\) and think they made an algebraic error, so they restart the problem using different approaches. They might try to solve each equation individually, finding that \(\mathrm{x^2 + y^2 = 9}\) gives infinitely many solutions (all points on that circle), and conclude the answer is "infinitely many." This may lead them to select Choice (D) (Infinitely many).

Second Most Common Error:

Conceptual confusion about circle equations: Students might not recognize these as circles and attempt to solve as a system of linear equations

They might try substitution or elimination methods that don't apply here, get confused by the identical left sides (\(\mathrm{x^2 + y^2}\)), and end up guessing. This leads to confusion and random answer selection.

The Bottom Line:

This problem tests whether students can recognize contradictions in algebraic manipulation and understand their geometric meaning. The key insight is that identical expressions (\(\mathrm{x^2 + y^2}\)) cannot simultaneously equal two different values (9 and 25).