Keeping an equation balanced: doing the same to both sides

Solving equations is one of the most useful skills in mathematics. It shows up in physics formulas, in working out prices and in almost every later topic in algebra. The good news is that nearly every method rests on one simple idea: balance.

An equation is a balanced scale

The equals sign means the left side and the right side have exactly the same value. Picture an old-fashioned balance scale with equal weights on each pan. If you remove 5 grams from one pan only, the scale tips. If you remove 5 grams from both pans, it stays level.

An equation works the same way. You may change it however you like, as long as you do exactly the same thing to both sides. Then the two sides stay equal, and the new equation has the same solution as the old one.

Undo operations in reverse order

Your goal is to get the unknown, usually x, on its own. To do that, undo whatever has been done to x, using inverse operations:

Each operation has an opposite that undoes it:

• Adding is undone by subtracting.
• Subtracting is undone by adding.
• Multiplying is undone by dividing.
• Dividing is undone by multiplying.

Work from the outside in. Usually that means dealing with any number added or subtracted first, then any number multiplying or dividing x.

When x appears on both sides

Sometimes the unknown appears on both sides, or inside brackets. The balance rule still applies. First expand any brackets, then gather the x terms on one side by adding or subtracting an x term from both sides.

Fractions are no different

If x is divided by a number, multiply both sides by that number to clear it.

Always check your answer

Checking takes seconds. Substitute your answer into the original equation and work out each side separately. If the two sides match, you are right. If not, look back through your steps for the one where the balance slipped.

Next time you solve an equation, write each step on a new line and say to yourself which operation you are doing to both sides. This habit will matter in the next lesson, where you will balance two equations at once.

Key takeaways

• An equation stays true only if you do the same thing to both sides.
• Use inverse operations to undo what has been done to x, working from the outside in.
• Check every answer by substituting it into the original equation.

Check your understanding

Question 1 of 3

Solve 5x - 7 = 18.

Choose an answer

x = 2.2x = 5x = 25x = 11/5

Check answer

Question 2 of 3

Which of these is a valid next step for 4x + 3 = 2x + 9?

Choose an answer

Subtract 2x from the left side onlyDivide only the 4x by 4Add 9 to the left side and subtract 9 from the right sideSubtract 3 from both sides

Check answer

Question 3 of 3

Solve x/3 + 1 = 5.

Choose an answer

x = 12x = 18x = 4/3x = 6

Check answer