10x + 3y = 1058x - 3y = -78The solution to the given system of equations is \((\mathrm{x}, \mathrm{y})\). What...

1. TRANSLATE the problem information

• Given information:
  • Equation 1: \(10\mathrm{x} + 3\mathrm{y} = 105\)
  • Equation 2: \(8\mathrm{x} - 3\mathrm{y} = -78\)
• We need to find the value of y in the solution \((\mathrm{x}, \mathrm{y})\)

2. INFER the most efficient solution method

• Look at the coefficients: \(+3\mathrm{y}\) in the first equation and \(-3\mathrm{y}\) in the second equation
• Since these are opposites, adding the equations will eliminate the y-variable completely
• This makes elimination by addition the fastest approach

3. SIMPLIFY by adding the equations

• Add the left sides and right sides:

\((10\mathrm{x} + 3\mathrm{y}) + (8\mathrm{x} - 3\mathrm{y}) = 105 + (-78)\)

• The y-terms cancel:

\(10\mathrm{x} + 8\mathrm{x} + 3\mathrm{y} + (-3\mathrm{y}) = 27\)

• This gives us:

\(18\mathrm{x} = 27\)

4. SIMPLIFY to solve for x

• Divide both sides by 18:

\(\mathrm{x} = \frac{27}{18}\)

• Reduce the fraction:

\(\mathrm{x} = \frac{3}{2}\)

5. SIMPLIFY by substituting to find y

• Use the first equation:

\(10\mathrm{x} + 3\mathrm{y} = 105\)

• Substitute \(\mathrm{x} = \frac{3}{2}\):

\(10\left(\frac{3}{2}\right) + 3\mathrm{y} = 105\)

• Multiply:

\(15 + 3\mathrm{y} = 105\)

• Subtract 15:

\(3\mathrm{y} = 90\)

• Divide by 3:

\(\mathrm{y} = 30\)

Answer: 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the elimination opportunity and choosing substitution instead, leading to more complex fractions and higher chance of arithmetic errors.

Students might solve the first equation for x: \(\mathrm{x} = \frac{105 - 3\mathrm{y}}{10}\), then substitute this into the second equation, creating: \(8\left[\frac{105 - 3\mathrm{y}}{10}\right] - 3\mathrm{y} = -78\). This leads to complex fraction arithmetic that's more error-prone than the simple addition approach.

This doesn't necessarily lead to a wrong final answer, but makes the problem much harder than it needs to be.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic mistakes when reducing \(\frac{27}{18}\) to \(\frac{3}{2}\), or errors in the substitution step.

For example, some students might incorrectly simplify \(\frac{27}{18}\) as \(\frac{9}{6}\) instead of \(\frac{3}{2}\), or make sign errors when calculating \(10\left(\frac{3}{2}\right) = 15\). These arithmetic errors propagate through to the final answer.

This leads to confusion and incorrect answer selection.

The Bottom Line:

The key insight is recognizing when the problem structure (opposite coefficients) makes one method much more efficient than others. Students who miss this strategic insight often still get the right answer, but work much harder than necessary.