x + y = 10 x - y = 14 The solution to the given system of equations is \(\mathrm{(x,...

1. INFER the best solution strategy

• Given information:
  • \(\mathrm{x + y = 10}\) (equation 1)
  • \(\mathrm{x - y = 14}\) (equation 2)
  • Need to find: value of y

• Key insight: Notice that the coefficients of y are +1 and -1, which means adding the equations will eliminate y entirely, or subtracting will eliminate x entirely.

2. INFER which elimination approach to use

• We can either:
  • Add the equations to eliminate y and solve for x first
  • Subtract equation (2) from (1) to eliminate x and solve directly for y

• Let's try the subtraction method since it gets us y directly.

3. SIMPLIFY by subtracting the equations

• Subtract equation (2) from equation (1):
  \(\mathrm{(x + y) - (x - y) = 10 - 14}\)
• Distribute the negative sign:
  \(\mathrm{x + y - x + y = -4}\)
• Combine like terms:
  \(\mathrm{2y = -4}\)
• Solve for y:
  \(\mathrm{y = -2}\)

4. Verify using the addition method

• Add both equations:
  \(\mathrm{(x + y) + (x - y) = 10 + 14}\)
  \(\mathrm{2x = 24}\)
  \(\mathrm{x = 12}\)
• SIMPLIFY by substituting back:
  \(\mathrm{12 + y = 10}\)
  \(\mathrm{y = -2}\) ✓

Answer: (A) -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elimination strategy and try to solve each equation individually, not understanding that they need both equations working together to find unique values.

This leads to confusion about how to proceed and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when subtracting equations, particularly with the negative sign distribution in \(\mathrm{(x + y) - (x - y)}\). They might get:
x + y - x - y = -4, which gives \(\mathrm{0 = -4}\) (impossible)

This causes them to think there's no solution and leads to random answer selection.

The Bottom Line:

Systems of equations require recognizing that you can manipulate entire equations just like individual terms - the key insight is seeing which operations will eliminate variables efficiently.