How many solutions does the equation \(5(3\mathrm{x} - 2) + \mathrm{x} = 4(4\mathrm{x} - 2.5)\) have?Exactly oneExactly twoInfinitely manyZero

1. SIMPLIFY the left side of the equation

• Expand using distributive property: \(5(3\mathrm{x} - 2) + \mathrm{x}\)
• \(5(3\mathrm{x} - 2) = 15\mathrm{x} - 10\)
• Add the remaining x term: \(15\mathrm{x} - 10 + \mathrm{x} = 16\mathrm{x} - 10\)

2. SIMPLIFY the right side of the equation

• Expand using distributive property: \(4(4\mathrm{x} - 2.5)\)
• \(4(4\mathrm{x} - 2.5) = 16\mathrm{x} - 10\)

3. SIMPLIFY by setting both sides equal and solving

• We now have: \(16\mathrm{x} - 10 = 16\mathrm{x} - 10\)
• Subtract 16x from both sides: \(16\mathrm{x} - 10 - 16\mathrm{x} = 16\mathrm{x} - 10 - 16\mathrm{x}\)
• This gives us: \(-10 = -10\)

4. INFER what this result means

• The equation \(-10 = -10\) is always true (it's an identity)
• When we get \(0 = 0\) or any true statement like this, it means the original equation is true for ALL values of x
• Therefore, the equation has infinitely many solutions

Answer: C - Infinitely many

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Misinterpreting what \(0 = 0\) means

Students correctly expand and simplify to get \(-10 = -10\), but then think "this doesn't make sense" or "there's no solution because we lost the variable." They don't recognize that getting a true statement means the original equation works for any value of x.

This may lead them to select Choice D (Zero) or cause confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors during expansion

Students make mistakes like \(5(3\mathrm{x} - 2) = 15\mathrm{x} - 2\) (forgetting to distribute to both terms) or \(4(4\mathrm{x} - 2.5) = 16\mathrm{x} - 8\) (calculating \(4 \times 2.5\) incorrectly). These errors lead to different expressions on each side, making them think there's exactly one solution when they solve.

This may lead them to select Choice A (Exactly one).

The Bottom Line:

The key insight is recognizing that when both sides of an equation simplify to identical expressions, resulting in a true statement like \(0 = 0\), this indicates the equation is an identity with infinitely many solutions - not no solution or exactly one solution.