Linear Solutions Poster
Types of Linear Equation Solutions
Rules for Rate of Change (ROC) & Constant
1. One Solution
Intersecting Lines (1 Point)
Single Value: x = #
Core Rule
DIFFERENT
Rate of Change (ROC)
Constants: Do NOT matter!
Example A Diff ROC
4x + 3 = 2x + 11
• ROC: 4 ≠ 2 (Different!)
• 2x = 8 → x = 4
Example B Diff ROC
5x - 7 = -1x + 5
• ROC: 5 ≠ -1 (Different!)
• 6x = 12 → x = 2
2. No Solution (∅)
Parallel Lines (Never Touch)
False Statement: a = b
Core Rule
SAME ROC
DIFFERENT Constant
Variables cancel out!
Example A False Identity
3x + 4 = 3x - 5
• ROC: 3 = 3 (Same)
• Constant: 4 ≠ -5 (Diff)
• Result: 4 = -5 ✗ FALSE!
Example B Distribute 1st
2(x + 3) = 2x + 9
• Simplify: 2x + 6 = 2x + 9
• ROC: 2 = 2 | Const: 6 ≠ 9
• Result: 6 = 9 ✗ FALSE!
3. Infinitely Many Solutions (∞)
Same Line (Infinite Intersections)
True Statement: a = a
Core Rule
SAME ROC
SAME Constant
Exact Same Equation!
Example A All Real #s
5x + 8 = 5x + 8
• ROC: 5 = 5 (Same)
• Constant: 8 = 8 (Same)
• Result: 8 = 8 ✓ TRUE!
Example B Distribute 1st
4(2x - 1) = 8x - 4
• Simplify: 8x - 4 = 8x - 4
• ROC: 8 = 8 | Const: -4 = -4
• Result: -4 = -4 ✓ TRUE!
The 2-Step Decision Flow Always simplify both sides first before comparing!
Step 1: Check ROC
Are ROCs different?
YES → ONE SOLUTION
Step 2: Same ROC
Different Constants?
YES → NO SOLUTION
Step 2: Same ROC
Same Constants?
YES → INFINITE SOLUTIONS