Variables on both sides

The equation 3x + 2 = x + 10 has the mystery amount on both pans of the scale. To solve, gather all the x's in one place and all the numbers in the other. Every gather must be a balanced move, applied to both sides. The x-term on the right does not get to vanish; it has to be paid for by the same subtraction on the left.

Milo's mistake was treating the move as a transfer with a sign change, like a carrier moving 3x to the right and flipping it, or adding x to the left. A true equation never gets carried; it only gets the same operation on both sides. Adding x made the left side 4x while the balance was silently broken, and 3 did not check in the original equation.

The fix is subtracting x from both sides. The left becomes 3x - x = 2x, and the right becomes x - x + 10 = 10, leaving 2x + 2 = 10. Then subtract 2 from both sides: 2x = 8, and divide by 2: x = 4. It is the same two-step recipe as before, with one extra gather at the front.

Check in the original: 3 * 4 + 2 = 14 on the left and 4 + 10 = 14 on the right, so the equation is balanced. The check is the whole point: an equation stays useful only if the value makes both sides truly equal. If a move felt like magic, substitution is the referee that shows whether it worked.