Variables on Both Sides

5x + 2 = 3x + 10 has x on both sides, so neither side can be worked out yet.

Gather the variables

Subtract 3x from both sides. That leaves 2x + 2 = 10, which is a two-step equation.

Then it is familiar

Subtract 2, then divide by 2, giving x = 4.

Check on both sides

5 × 4 + 2 = 22, and 3 × 4 + 10 = 22. Both match, so it is right.

Collect the letters on one side

For 5x + 3 = 2x + 12, subtract 2x from both sides to get 3x + 3 = 12. Once the variable appears on one side only, the equation is an ordinary two-step problem. Gathering first is what reduces the new case to the known one.

Which side to gather on

Move the smaller coefficient to avoid negatives: with 5x and 2x, subtract the 2x. Either choice gives the same answer, but one leaves cleaner arithmetic. Small choices like this add up over a long calculation.

The balance principle is unchanged

Subtracting 2x from both sides is legitimate for exactly the same reason as subtracting 2 — you did the same thing to both sides. Nothing new is being assumed; the thing being subtracted just happens to contain a letter.

Substitute into the original

For x = 3: left side 5(3) + 3 = 18, right side 2(3) + 12 = 18. Both match. Checking both sides separately is the natural check when the variable started on both.