x = -4 y = 2x^2 - 10 The graphs of the equations in the given system of equations intersect...

1. TRANSLATE the problem setup

• Given information:
  • First equation: \(\mathrm{x = -4}\)
  • Second equation: \(\mathrm{y = 2x^2 - 10}\)
  • Need: The y-coordinate where these graphs intersect

• What this tells us: Since the intersection point must satisfy both equations, we already know \(\mathrm{x = -4}\), so we need to find the corresponding y-value.

2. TRANSLATE the solution approach

• The intersection point (x, y) must satisfy both equations
• Since we know \(\mathrm{x = -4}\) from the first equation, substitute this value into the second equation to find y

3. SIMPLIFY through substitution and calculation

• Substitute \(\mathrm{x = -4}\) into \(\mathrm{y = 2x^2 - 10}\):
  \(\mathrm{y = 2(-4)^2 - 10}\)

• Apply order of operations - exponents first:
  \(\mathrm{(-4)^2 = 16}\)

• Continue the calculation:
  \(\mathrm{y = 2(16) - 10}\)
  \(\mathrm{y = 32 - 10}\)
  \(\mathrm{y = 22}\)

Answer: C) 22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative exponents: Students calculate \(\mathrm{(-4)^2}\) as \(\mathrm{-16}\) instead of \(\mathrm{16}\).

They reason: "Since -4 is negative, \(\mathrm{(-4)^2}\) should be negative too." This fundamental misunderstanding of exponentiation leads to:
\(\mathrm{y = 2(-16) - 10 = -32 - 10 = -42}\)

This may lead them to select Choice A (-42).

Second Most Common Error:

Poor order of operations in SIMPLIFY: Students perform operations left to right instead of following PEMDAS.

They calculate: \(\mathrm{y = 2(-4)^2 - 10}\) as \(\mathrm{y = (2 \times -4)^2 - 10 = (-8)^2 - 10 = 64 - 10 = 54}\), or make similar sequencing errors that don't match any answer choice. This leads to confusion and guessing.

The Bottom Line:

This problem tests careful arithmetic execution more than complex mathematical reasoning. The key insight is that negative numbers squared become positive, and maintaining proper order of operations is crucial for accuracy.