Rocket slaps a puzzle card on the bench. "2x + 3 = 2x + 5. I have tried every number up to twenty. Nothing works."
"What do you notice if you take 2x off both sides?" Raven asks.
"3 = 5," Rocket says. "That is just false. Not a single x can fix it."
Raven turns over another card: 3(x + 2) = 3x + 6. "Try this one."
Rocket distributes. "3x + 6 = 3x + 6. Take 3x off both sides and I get 6 = 6. Always true."
Nova hovers between the cards, her light flicking from one to the other. "How many solutions is always true?"
"Every number," Rocket says. "Infinitely many. And the other card has none."
"So an equation ends three ways," Raven says. "x equals a number, a number equals itself, or two numbers disagree."
Simplify any linear equation and it ends one of three ways.
x = a: one solution. The letter stays, and one number works, such as x = 4.
a = a: infinitely many solutions. The letters cancel and the numbers agree, such as 6 = 6.
a = b: no solution. The letters cancel and the numbers disagree, such as 3 = 5.
| Ending | Example | Solutions |
|---|---|---|
| x = a | 4x - 1 = 2x + 7 becomes x = 4 | one |
| a = a | 3(x + 2) = 3x + 6 becomes 6 = 6 | infinitely many |
| a = b | 2x + 3 = 2x + 5 becomes 3 = 5 | none |
The crew sorted this week's puzzle cards by ending. These are the crew's own cards, written for the Puzzle Post.
| Card | Equation | After simplifying | Solutions |
|---|---|---|---|
| 1 | 4x - 1 = 2x + 7 | x = 4 | one solution |
| 2 | 2x + 3 = 2x + 5 | a = b | no solution |
| 3 | 3(x + 2) = 3x + 6 | a = a | infinitely many |
| 4 | 5x - 4 = 3x + 10 | x = 7 | one solution |
| 5 | 2(x + 4) = 2x + 8 | ? | ? |
| 6 | x + 9 = 3x - 1 | ? | ? |
| Statement | True or false? |
|---|---|
| 2x + 3 = 2x + 5 has no solution. | ? |
| 3(x + 2) = 3x + 6 has infinitely many solutions. | ? |
| 5x - 4 = 3x + 10 has infinitely many solutions. | ? |
| If the letters cancel and 7 = 7 is left, every number is a solution. | ? |
| An equation with no solution has x = 0 as its answer. | ? |
Sharp sorting. Tomorrow you meet equations like these in everyday life and review the whole week.