Wren writes 3(x + 2) = 21 on the whiteboard. "Two ways to solve this. Comet, you first."
"Distribute. 3x + 6 = 21. Subtract 6 from both sides, 3x = 15. Divide by 3, x = 5."
"Now mine," Wren says. "Divide both sides by 3 first. x + 2 = 7. Subtract 2. x = 5."
"Same answer," Comet says. "Yours was shorter. What do you notice?"
"Both work because every step kept the two sides equal," Wren says. "If a = b, then a ÷ 3 = b ÷ 3."
Nova hovers between the two columns, her light tracing each line. "Would you like a hint? Say the reason out loud."
"Equal things, treated the same, stay equal," Comet says. "That is the whole game."
Start by assuming the equation has a solution. Then both sides name the same number.
If two numbers are equal, adding the same amount to each keeps them equal. So does subtracting, multiplying or dividing by a nonzero number.
Every solving step is one of those moves. That is why the new equation has the same solution as the old one.
| Move | Example | Reason |
|---|---|---|
| Subtract the same number from both sides | 3x + 6 = 21 becomes 3x = 15 | Equal numbers minus 6 are still equal. |
| Divide both sides by the same nonzero number | 3x = 15 becomes x = 5 | Equal numbers divided by 3 are still equal. |
| Distribute | 3(x + 2) becomes 3x + 6 | The two expressions name the same number for every x. |
| Collect like terms | 2x + 9 = 5x - 3 becomes 9 = 3x - 3 | Subtracting 2x from both sides keeps them equal. |
Comet distributed first: 3x + 6 = 21, then 3x = 15, then x = 5.
Wren divided first: x + 2 = 7, then x = 5.
Both are correct. Dividing first is shorter here because 21 divides evenly by 3. Pick the path with less work.
For 2x + 9 = 5x - 3, subtract 2x from both sides: 9 = 3x - 3. Add 3: 12 = 3x. Divide: x = 4.
Sometimes the letters cancel completely. If what is left is true, like 2 = 2, every value works. If it is false, like 2 = 5, nothing works.
| Statement | True or false? |
|---|---|
| If a = b, then a - 6 = b - 6. | ? |
| Distributing first and dividing first can both solve 3(x + 2) = 21. | ? |
| x = 4 is a solution of 2x + 9 = 5x - 3. | ? |
| x = 7 is a solution of 3(x + 2) = 21. | ? |
| If the letters cancel and 2 = 5 is left, every value is a solution. | ? |
Excellent reasoning. Tomorrow is Loft Lab: you build the coat-hanger balance and solve with real beans.