10 = 2x + 4How many solutions exist to the equation shown above?

1. TRANSLATE the problem information

• Given equation: \(10 = 2\mathrm{x} + 4\)
• Question asks: How many solutions exist?
• What this tells us: We need to solve for x, then determine how many values of x satisfy the equation

2. INFER the approach

• This is a linear equation in one variable
• Strategy: Use inverse operations to isolate x
• The number of solutions will depend on what we get when we solve

3. SIMPLIFY by eliminating the constant term

• Subtract 4 from both sides: \(10 - 4 = 2\mathrm{x} + 4 - 4\)
• This gives us: \(6 = 2\mathrm{x}\)

4. SIMPLIFY by isolating the variable

• Divide both sides by 2: \(\frac{6}{2} = \frac{2\mathrm{x}}{2}\)
• This gives us: \(3 = \mathrm{x}\)

5. INFER the final answer

• We found exactly one value: \(\mathrm{x} = 3\)
• We can verify: \(10 = 2(3) + 4 = 6 + 4 = 10\) ✓
• Since we have exactly one value that works, there is exactly one solution

Answer: B. Exactly 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about solution types: Students may think that because solving the equation takes multiple steps, there might be multiple solutions, or they may confuse this with quadratic equations that can have multiple solutions.

This conceptual gap leads them to select Choice C (Exactly 3) or guess between the options rather than recognizing that linear equations in one variable have exactly one solution when they simplify to \(\mathrm{x} = \text{[some number]}\).

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors during the solving process, such as incorrectly subtracting 4 or dividing by 2, leading to wrong values for x.

While this doesn't necessarily change the number of solutions, it can cause confusion about whether their answer is correct, leading to second-guessing and potentially selecting Choice A (None) if they think their work must be wrong.

The Bottom Line:

This problem tests both algebraic manipulation skills and conceptual understanding of what different solution types mean. The key insight is recognizing that when a linear equation simplifies to "\(\mathrm{x} = \text{[specific number]}\)," there is exactly one solution, regardless of how many steps it took to get there.