The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of x?x^2 + y +...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x^2 + y + 10 = 10}\)
  • \(\mathrm{8x + 16 - y = 0}\)
• We need to find the value of x

2. INFER the best solution strategy

• Since one equation is linear in y, substitution will be most efficient
• We should solve the simpler equation for y, then substitute into the quadratic equation

3. SIMPLIFY each equation to isolate y

• First equation: \(\mathrm{x^2 + y + 10 = 10}\) → \(\mathrm{x^2 + y = 0}\) → \(\mathrm{y = -x^2}\)
• Second equation: \(\mathrm{8x + 16 - y = 0}\) → \(\mathrm{y = 8x + 16}\)

4. INFER that both expressions equal y, so we can set them equal

• Since \(\mathrm{y = -x^2}\) and \(\mathrm{y = 8x + 16}\):
• \(\mathrm{-x^2 = 8x + 16}\)

5. SIMPLIFY to get standard quadratic form

• \(\mathrm{-x^2 = 8x + 16}\)
• \(\mathrm{-x^2 - 8x - 16 = 0}\)
• \(\mathrm{x^2 + 8x + 16 = 0}\)

6. INFER that this is a perfect square trinomial

• Notice that \(\mathrm{x^2 + 8x + 16 = (x + 4)^2}\)
• So \(\mathrm{(x + 4)^2 = 0}\)

7. SIMPLIFY to find the solution

• Taking the square root: \(\mathrm{x + 4 = 0}\)
• Therefore: \(\mathrm{x = -4}\)

Answer: B. -4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students solve correctly but find the value of y instead of x. Since the problem asks "What is the value of x?" they might substitute \(\mathrm{x = -4}\) back into \(\mathrm{y = -x^2}\) to get \(\mathrm{y = -16}\), then incorrectly select this as their answer.

This leads them to select Choice A (-16).

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors during the substitution step. When setting \(\mathrm{-x^2 = 8x + 16}\), they might incorrectly rearrange as \(\mathrm{x^2 - 8x - 16 = 0}\) instead of \(\mathrm{x^2 + 8x + 16 = 0}\). This leads to a different quadratic that doesn't factor as nicely.

This causes confusion and may lead to guessing among the remaining choices.

The Bottom Line:

This problem requires careful attention to what variable you're solving for and systematic algebraic manipulation. The substitution method makes it straightforward, but sign errors and misreading the question are the main pitfalls.