Variables on both sides

The question

10x + 13 = 12x + 3

10x + 13=12x + 3

10x + 13 = 12x + 3

13 = 2x + 3

2x + 3 = 13

2x + 3=13

2x=10

x=5

10x + 13 = 12x + 3

63 = 63✓

This one has x on both sides, so collect the x terms first.
Take 10x off both sides. That clears x from the left, leaving 13 = 2x + 3. Swap the sides so x is on the left: 2x + 3 = 13.
Two things were done to x: multiplied by 2, then 3 added. Undo them in reverse order — whatever happened last comes off first.
The 3 went on last, so it comes off first — from both sides, to keep them equal. 13 − 3 = 10.
That leaves 2x, so divide both sides by 2: 10 ÷ 2 = 5.
Check by putting it back into the original, both sides: 10(5) + 13 = 63, and 12(5) + 3 = 63 — both 63, so x = 5.

How it works.

01
This one has x on both sides, so collect the x terms first.
02
Take 10x off both sides. That clears x from the left, leaving 13 = 2x + 3. Swap the sides so x is on the left: 2x + 3 = 13.
03
Two things were done to x: multiplied by 2, then 3 added. Undo them in reverse order — whatever happened last comes off first.
04
The 3 went on last, so it comes off first — from both sides, to keep them equal. 13 − 3 = 10.
05
That leaves 2x, so divide both sides by 2: 10 ÷ 2 = 5.
06
Check by putting it back into the original, both sides: 10(5) + 13 = 63, and 12(5) + 3 = 63 — both 63, so x = 5.