Isolating the Variable in Any Linear Equation

A routine that always finishes

Real equations rarely arrive in the friendly form 3x + 5 = 20. They have variables on both sides, parentheses, and fractions. The good news: one fixed routine handles all of them. The goal never changes — isolate the variable so the last line reads x = some number.

1. Clear fractions by multiplying every term on both sides by the least common denominator.
2. Remove parentheses with the distributive property.
3. Combine like terms on each side so each side is as simple as possible.
4. Collect all variable terms on one side and all constants on the other.
5. Divide both sides by the variable's coefficient, then check in the original.

Variables on both sides

When the unknown appears on both sides, gather it on whichever side keeps its coefficient positive — that small choice spares you a sign error later. Then finish as usual.

Solve   5x - 4 = 2x + 11

Move 2x to the left (subtract 2x from both sides):
  5x - 2x - 4 = 11
  3x - 4 = 11
Move the -4 (add 4 to both sides):
  3x = 15
Divide by 3:
  x = 5

Check: left 5(5) - 4 = 21; right 2(5) + 11 = 21.  21 = 21  TRUE.

Parentheses and fractions

Parentheses come off with the distributive property: 2(x - 3) = 2x - 6. Be careful when a minus sign sits in front — -(x - 3) flips both signs to -x + 3. Fractions are best removed up front by clearing denominators, multiplying every term by the common denominator so you can work with whole numbers.

Solve   (x/2) + (x/3) = 5

LCD of 2 and 3 is 6. Multiply EVERY term by 6:
  6*(x/2) + 6*(x/3) = 6*5
  3x + 2x = 30
Combine like terms:
  5x = 30
Divide by 5:
  x = 6

Check: 6/2 + 6/3 = 3 + 2 = 5  TRUE.