Equations with No Solution, One Solution, or Infinitely Many Solutions

When you solve a linear equation, one of three things can happen: 1. One solution — You get a statement like x = 5. Most equations work this way. 2. No solution — You get a false statement like 0 = 7 or 3 = -1. This means no value of x can ever make the equation true. We call this a contradiction. 3. Infinitely many solutions — You get a true statement like 0 = 0 or 5 = 5. This means every val…

After simplifying, look at what happens to the x-terms: | What you get | Meaning | Example | |---|---|---| | x = some number | One solution | 3x = 12 → x = 4 | | A false number statement | No solution | 0 = 7 | | A true number statement | Infinitely many | 0 = 0 | The trick: if the x-terms cancel out on both sides, you're left with either a contradiction or an identity.

Example: How many solutions does 2(x + 3) = 2x + 6 have?