\(8(\mathrm{y} + 3) = 24 + 8\mathrm{y}\) How many solutions does the given equation have?...

1. SIMPLIFY the left side using distribution

• Given equation: \(8(\mathrm{y} + 3) = 24 + 8\mathrm{y}\)
• Apply distributive property to the left side:
  • \(8(\mathrm{y} + 3) = 8\mathrm{y} + 24\)
• Rewritten equation: \(8\mathrm{y} + 24 = 24 + 8\mathrm{y}\)

2. SIMPLIFY by isolating terms

• Subtract 8y from both sides:
  • \(8\mathrm{y} + 24 - 8\mathrm{y} = 24 + 8\mathrm{y} - 8\mathrm{y}\)
  • \(24 = 24\)

3. INFER what this result means

• When an equation reduces to a statement that's always true (like \(24 = 24\)), this means:
  • The original equation is satisfied by ALL possible values of y
  • We call this an "identity"
  • The equation has infinitely many solutions

Answer: C) Infinitely many

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students see that the variable "disappeared" and think this means there's no solution.

When they get \(24 = 24\), they reason: "The variable y is gone, so there's no way to solve for y. This must mean no solutions exist." This fundamental misunderstanding of what an identity represents may lead them to select Choice D (Zero).

Second Most Common Error:

Incomplete SIMPLIFY execution: Students make algebraic errors during the distribution or equation manipulation steps.

For example, they might incorrectly distribute to get \(8\mathrm{y} + 3\) instead of \(8\mathrm{y} + 24\), or make sign errors when moving terms. These calculation mistakes lead to different equations that don't reduce to an identity, causing them to get stuck and guess randomly among the choices.

The Bottom Line:

This problem tests whether students understand the deeper meaning behind algebraic manipulation. The key insight is recognizing that when an equation reduces to a true statement (rather than something like \(\mathrm{y} = 5\)), it means every possible value works as a solution.