Linear Solutions PosterLinear Equation Solutions Key Strategy Compare ROC & Constant Simplified Form: ax + b = cx + d Rate of Change (ROC) Constants 1 One Solution Single Value: x = # The Rule DIFFERENT ROC Lines cross at 1 point (a ≠ c) Example ROC: 4 ≠ 2 4x + 3 = 2x + 11 ✓ One Solution: x = 4 2 No Solution (∅) False Statement: a = b The Rule SAME ROC DIFF Constants Parallel lines that never touch Example ROC: 3 = 3 | Const: 4 ≠ -5 3x + 4 = 3x - 5 ✗ 4 = -5 (FALSE STATEMENT) 3 Infinite Solutions (∞) True Statement: a = a The Rule SAME ROC SAME Constants Same line • Infinite touches Example ROC: 5 = 5 | Const: 8 = 8 5x + 8 = 5x + 8 ✓ 8 = 8 (ALWAYS TRUE) Quick 2-Step Decision Flow Always simplify both sides first! Step 1: Check ROC Are ROCs DIFFERENT? YES → ONE SOLUTION Step 2: Same ROC Constants DIFFERENT? YES → NO SOLUTION Step 2: Same ROC Constants the SAME? YES → INFINITE (∞)
