The Balancing Principle

Think of an equation like a pair of scales. The equals sign is the pivot in the middle. If you throw a weight on one side, the scales tip. So you have to add the same weight to the other side to keep things level. That's all you're doing when you solve an equation.

  • Add the same number to both sides and it stays balanced.
  • Subtract the same number from both sides and it stays balanced.
  • Multiply or divide both sides by the same non-zero number and it stays balanced.

The goal is to get the variable alone. One on its own, with no coefficient hanging around it. You do that by applying the opposite of whatever's being done to it. Addition is undone by subtraction. Multiplication is undone by division.

Core rule: if A = B, then A + k = B + k for any value k
Also: if A = B, then A × k = B × k (provided k ≠ 0)

One-Step Equations

These are the simplest kind. One move and you're done. Just figure out what's happening to the variable and do the opposite to both sides.

Example A - addition: x + 9 = 15
Subtract 9 from both sides: x = 15 − 9 = 6

Example B - multiplication: 7n = 42
Divide both sides by 7: n = 42 ÷ 7 = 6

Example C - division: y/4 = 3
Multiply both sides by 4: y = 12

Always check by plugging your answer back into the original equation. For Example A: 6 + 9 = 15. Yep, that works.

Two-Step Equations

Now you've got two things to undo. Here's the order that works: deal with the addition or subtraction first, then handle the multiplication or division. It's the reverse of how the equation was built.

Solve: 3x + 4 = 19

Step 1 - Subtract 4 from both sides:
3x = 15

Step 2 - Divide both sides by 3:
x = 5

Check: 3(5) + 4 = 15 + 4 = 19 ✓
Solve: (y − 2) / 5 = 4

Step 1 - Multiply both sides by 5:
y − 2 = 20

Step 2 - Add 2 to both sides:
y = 22

Check: (22 − 2) / 5 = 20 / 5 = 4 ✓

Variables on Both Sides

Okay, so here's where people get confused. When the variable shows up on both sides, you can't just start isolating it right away. First, get all the variable terms onto one side and all the numbers onto the other. Pick the side that gives you a positive x coefficient, it'll save you from sign errors down the line.

Solve: 5x − 3 = 2x + 9

Step 1 - Subtract 2x from both sides:
3x − 3 = 9

Step 2 - Add 3 to both sides:
3x = 12

Step 3 - Divide by 3:
x = 4

Check: 5(4) − 3 = 17 and 2(4) + 9 = 17 ✓

Sometimes the x-terms cancel out completely. If what's left is something true like 0 = 0, then every number is a solution. If you land on something impossible like 3 = 7, there's no solution at all.

Word Problems into Equations

Word problems have a bad reputation, but they're manageable once you have a system. Here's what to do:

  1. Figure out what you don't know and give it a letter. Usually x works fine.
  2. Look for the key words and translate them: "more than" means add, "times as many" means multiply, "split equally" means divide.
  3. Write the equation based on the relationship the problem describes.
  4. Solve it, then go back and re-read the question. Make sure you answered what was actually asked, not just whatever x turned out to be.
Problem: A bus carries 18 more passengers than a car. Together they carry 42 people. How many does the car carry?

Let c = number of passengers in the car
Bus carries: c + 18
Together: c + (c + 18) = 42

Simplify: 2c + 18 = 42
Subtract 18: 2c = 24
Divide by 2: c = 12

The car carries 12 passengers; the bus carries 30.

That last step really matters. Go back and read what they asked. In this example, if the problem wanted the bus count, you'd need c + 18, not just c. Easy point to lose if you rush through it.