-{x + y = -3.5} x + 3y = 9.5 If \((\mathrm{x}, \mathrm{y})\) satisfies the system of equations above, what...

1. TRANSLATE the problem information

• Given information:
  • Equation 1: \(-\mathrm{x} + \mathrm{y} = -3.5\)
  • Equation 2: \(\mathrm{x} + 3\mathrm{y} = 9.5\)
• Find: the value of y

2. INFER the most efficient approach

• Looking at the coefficients: we have \(-\mathrm{x}\) in the first equation and \(+\mathrm{x}\) in the second equation
• These will cancel out perfectly if we add the equations together
• This elimination approach will give us an equation with only y

3. SIMPLIFY by adding the equations

• Add left sides: \((-\mathrm{x} + \mathrm{y}) + (\mathrm{x} + 3\mathrm{y}) = -\mathrm{x} + \mathrm{y} + \mathrm{x} + 3\mathrm{y} = 4\mathrm{y}\)
• Add right sides: \(-3.5 + 9.5 = 6\)
• Combined equation: \(4\mathrm{y} = 6\)

4. SIMPLIFY to solve for y

• Divide both sides by 4: \(\mathrm{y} = 6/4 = 3/2\)

Answer: 3/2 (or 1.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elimination opportunity and instead attempt substitution method, leading to more complex algebraic manipulation. They might solve the first equation for x (\(\mathrm{x} = \mathrm{y} + 3.5\)), substitute into the second equation, and make algebraic errors during the substitution process. This leads to confusion and potential arithmetic mistakes.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the elimination strategy but make arithmetic errors when combining like terms. They might incorrectly combine \(\mathrm{y} + 3\mathrm{y} = 3\mathrm{y}\) instead of \(4\mathrm{y}\), or make errors with the negative signs when adding \(-\mathrm{x} + \mathrm{x}\). These calculation errors lead to wrong values for y.

The Bottom Line:

Success on this problem depends on recognizing that the coefficients of x are opposites (\(-1\) and \(+1\)), making elimination the most direct method. Students who miss this insight often choose more complicated approaches that increase their chance of making errors.