5x - 3y = 8y + 1ry = -1/2 + 5/2xIn the given system of equations, r is a constant....

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{5x - 3y = 8y + 1}\)
  • \(\mathrm{ry = -\frac{1}{2} + \frac{5}{2}x}\)
• Find: value of r that makes the system have infinitely many solutions

2. SIMPLIFY both equations into standard form

• Equation 1: \(\mathrm{5x - 3y = 8y + 1}\)
  • Move all y terms to left: \(\mathrm{5x - 3y - 8y = 1}\)
  • Combine: \(\mathrm{5x - 11y = 1}\)
• Equation 2: \(\mathrm{ry = -\frac{1}{2} + \frac{5}{2}x}\)
  • Move x term to left: \(\mathrm{ry - \frac{5}{2}x = -\frac{1}{2}}\)
  • Rearrange: \(\mathrm{-\frac{5}{2}x + ry = -\frac{1}{2}}\)

3. INFER the condition for infinitely many solutions

• For a system to have infinitely many solutions, the equations must be equivalent
• This means one equation is a scalar multiple of the other
• Therefore, the ratios of corresponding coefficients must be equal:
  
  coefficient of x in eq1 / coefficient of x in eq2 = coefficient of y in eq1 / coefficient of y in eq2 = constant term in eq1 / constant term in eq2

4. SIMPLIFY to find the coefficient ratios

• Set up the ratios: \(\mathrm{\frac{5}{-\frac{5}{2}} = \frac{-11}{r} = \frac{1}{-\frac{1}{2}}}\)
• Calculate known ratios:
  • \(\mathrm{\frac{5}{-\frac{5}{2}} = 5 \times \frac{-2}{5} = -2}\)
  • \(\mathrm{\frac{1}{-\frac{1}{2}} = 1 \times (-2) = -2}\)
• Both equal -2, so: \(\mathrm{\frac{-11}{r} = -2}\)

5. SIMPLIFY to solve for r

• From \(\mathrm{\frac{-11}{r} = -2}\):
  • Cross multiply: \(\mathrm{-11 = -2r}\)
  • Divide both sides by -2: \(\mathrm{r = \frac{11}{2}}\)

Answer: \(\mathrm{\frac{11}{2}}\) (also acceptable: 5.5 or \(\mathrm{\frac{22}{4}}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "infinitely many solutions" means the equations must be equivalent. Instead, they might try to solve the system by substitution or elimination, leading to confusion when they encounter the parameter r. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating coefficient ratios, especially with the fraction division \(\mathrm{5 \div (-\frac{5}{2})}\). Common mistakes include getting positive 2 instead of negative 2, or incorrectly handling the fraction arithmetic. This may lead them to calculate \(\mathrm{r = -\frac{11}{2}}\) instead of \(\mathrm{r = \frac{11}{2}}\).

The Bottom Line:

This problem tests whether students understand what "infinitely many solutions" actually means for a system of equations - it's not just about solving, but about recognizing when two equations represent the same line.