Special Cases - No Solution and Infinite Solutions

Pre-Algebra · Axiom Academy

LESSON Special Cases in Equations When the variable terms cancel, an equation can have no solution at all — or infinitely many. 1. The Normal Case: One Solution Start with a typical equation so the special cases stand out by contrast. The two sides have different amounts of x ( 3x versus 2x ), so when we subtract 2x from both sides the variable does not vanish — one x survives, and we can solve for it. 2. No Solution: A False Statement Now make both sides carry the same variable term, 2x , but with different constants. Subtract 2x from both sides and the x cancels completely — there's no variable left to solve for. Watch what the leftover says. 3. Infinite Solutions: A True Statement Keep the same variable term on both sides, 2x , but now make the constants match too. The x cancels completely again — only this time the leftover is something that's always true. The whole story comes down to what the x terms do and, if they cancel, what's left behind . Here are all three side by side. 5. How to Identify a Special Case Turn the comparison into a quick decision you can run on any linear equation. The animation traces the path; the guide and practice below put it to work. Combine like terms and try to isolate the variable. Then check what happens to the variable: It survives (like x = 3 ) → One Solution It cancels and leaves a false statement (like 3 = 5 ) → No Solution It cancels and leaves a true statement (like 3 = 3 ) → Infinite Solutions

This is the written version of the interactive lesson above. See the full Pre-Algebra course.