30x - 30y = 10y + 20ry = 1/2 - 5xIn the given system of equations, r is a constant....

1. SIMPLIFY the given equations to standard form

• First equation: \(30\mathrm{x} - 30\mathrm{y} = 10\mathrm{y} + 20\)
  • Collect y terms: \(30\mathrm{x} - 30\mathrm{y} - 10\mathrm{y} = 20\)
  • Simplify: \(30\mathrm{x} - 40\mathrm{y} = 20\)
  • Divide by 10: \(3\mathrm{x} - 4\mathrm{y} = 2\)

• Second equation: \(\mathrm{ry} = \frac{1}{2} - 5\mathrm{x}\)
  • Rearrange: \(\mathrm{ry} + 5\mathrm{x} = \frac{1}{2}\)
  • Standard form: \(5\mathrm{x} + \mathrm{ry} = \frac{1}{2}\)

2. INFER the condition for no solution

• For a system \(\mathrm{ax} + \mathrm{by} = \mathrm{c}\) and \(\mathrm{dx} + \mathrm{ey} = \mathrm{f}\) to have no solution:
  • Coefficient ratios must be equal: \(\frac{\mathrm{a}}{\mathrm{d}} = \frac{\mathrm{b}}{\mathrm{e}}\)
  • But constant ratios must be different: \(\frac{\mathrm{a}}{\mathrm{d}} \neq \frac{\mathrm{c}}{\mathrm{f}}\)

• Our system: \(3\mathrm{x} + (-4)\mathrm{y} = 2\) and \(5\mathrm{x} + \mathrm{ry} = \frac{1}{2}\)
  • Need: \(\frac{3}{5} = \frac{-4}{\mathrm{r}} \neq \frac{2}{\frac{1}{2}}\)

3. SIMPLIFY to solve for r

• Solve the proportion: \(\frac{3}{5} = \frac{-4}{\mathrm{r}}\)
  • Cross multiply: \(3\mathrm{r} = 5(-4)\)
  • Calculate: \(3\mathrm{r} = -20\)
  • Divide: \(\mathrm{r} = \frac{-20}{3}\)

4. INFER verification of the no-solution condition

• Check that \(\frac{3}{5} \neq \frac{2}{\frac{1}{2}}\):
  • \(\frac{3}{5} = 0.6\)
  • \(\frac{2}{\frac{1}{2}} = 2 \times 2 = 4\)
  • Since \(0.6 \neq 4\), condition satisfied ✓

Answer: \(\frac{-20}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what 'no solution' means for a system of equations.

Students might try to solve the system directly by substitution or elimination, getting confused when they encounter a contradiction like \(0 = 5\). They don't realize they need to work backwards from the condition that creates this contradiction - when coefficient ratios are equal but constant ratios aren't. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when converting equations or solving the proportion.

Common mistakes include: forgetting to move all y terms when simplifying the first equation (getting \(30\mathrm{x} - 20\mathrm{y} = 20\) instead of \(30\mathrm{x} - 40\mathrm{y} = 20\)), or making sign errors when cross-multiplying \(\frac{3}{5} = \frac{-4}{\mathrm{r}}\). These errors lead to wrong values of r that don't actually create a no-solution system.

The Bottom Line:

This problem requires recognizing that 'no solution' is a specific mathematical condition, not just something that happens when you can't solve equations. The key insight is working with coefficient relationships rather than trying to find x and y values.