Which of the following points (x, y) is the solution to the given system of equations in the xy-plane? x...

1. TRANSLATE the problem requirements

• We need to find the point (x, y) that satisfies both equations simultaneously
• Given system:
  • \(\mathrm{x = 5}\)
  • \(\mathrm{y = x - 8}\)

2. INFER the solving strategy

• Since the first equation directly gives us \(\mathrm{x = 5}\), we can substitute this value into the second equation
• This substitution method will give us the y-coordinate

3. SIMPLIFY by substitution

• Substitute \(\mathrm{x = 5}\) into \(\mathrm{y = x - 8}\):
  \(\mathrm{y = 5 - 8}\)
  \(\mathrm{y = -3}\)
• The solution point is \(\mathrm{(5, -3)}\)

4. Verify against answer choices

• Choice B: \(\mathrm{(5, -3)}\) matches our solution

Answer: B. \(\mathrm{(5, -3)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making an arithmetic sign error when calculating \(\mathrm{5 - 8}\)

Students might think: "5 - 8... that's 3, right?" and forget about the negative sign, getting \(\mathrm{y = 3}\) instead of \(\mathrm{y = -3}\).

This may lead them to select Choice D: \(\mathrm{(5, 8)}\) or cause confusion since \(\mathrm{(5, 3)}\) isn't among the choices.

Second Most Common Error:

Poor INFER reasoning: Not recognizing that substitution is the key strategy

Some students see \(\mathrm{x = 5}\) and \(\mathrm{y = x - 8}\) as separate, unconnected pieces of information. They might try to check each answer choice individually rather than using the systematic substitution approach.

This leads to inefficient guessing or selecting Choice C: \(\mathrm{(5, -8)}\) by misreading the second equation as "\(\mathrm{y = -8}\)" instead of "\(\mathrm{y = x - 8}\)."

The Bottom Line:

This problem tests whether students can connect the two equations through substitution and perform careful arithmetic with negative numbers. The key insight is recognizing that having \(\mathrm{x = 5}\) immediately unlocks the value of y.