Raven writes a puzzle card and tapes it to the balance. "2(x + 3) = 4x - 2. No cups this time."
"Parentheses," Rocket says. "Two groups of x plus 3. That is 2x + 6 on the left."
"What do you notice about the right side?" Raven asks.
"4x take away 2. More x on the right than the left." Rocket frowns. "Which side do I clear?"
Nova hovers beside the card, her light steady. "Does it matter which side the x lands on?" she asks.
"No," Rocket says slowly. "I just want all the x on one side and all the numbers on the other."
He subtracts 2x from both sides, then adds 2. "6 + 2 = 2x. So x is 4."
"Check it," Raven says. Rocket does. Both sides make 14. He grins.
A linear equation only multiplies the letter by numbers and adds numbers. Every one can be solved with the same five moves.
The moves keep both sides equal, so the solution never changes. Only the look of the equation changes.
| Move | What it does | Example |
|---|---|---|
| Distribute | Multiply every term inside the parentheses | 2(x + 3) becomes 2x + 6 |
| Collect like terms | Add the x terms together, and the plain numbers together | 3x + 2x + 1 becomes 5x + 1 |
| Letters to one side | Add or subtract the same x term on both sides | 2x + 6 = 4x - 2 becomes 6 = 2x - 2 |
| Numbers to the other side | Add or subtract the same number on both sides | 6 = 2x - 2 becomes 8 = 2x |
| Divide | Divide both sides by the number in front of x | 8 = 2x becomes x = 4 |
Check: 2(4 + 3) = 14 and 4 × 4 - 2 = 14. Both sides agree, so x = 4.
Not every equation needs every move. Use the ones the equation calls for, in this order.
A fraction or decimal coefficient is handled the same way. For 1/2x + 3 = 7, subtract 3: 1/2x = 4. Then divide by 1/2, which is the same as multiplying by 2: x = 8.
For 0.5x + 1.5 = 4, subtract 1.5 to get 0.5x = 2.5. Divide by 0.5: x = 5.
You can also clear the fraction first. Multiply every term of 1/2x + 3 = 7 by 2 to get x + 6 = 14, so x = 8.
Excellent method work. Tomorrow you load real envelopes on both cups in the Puzzle Lab.