Can I do absolutely any operation to both sides?

Almost anything. You can add, subtract, multiply, or divide by the same number. You can even square both sides or take the square root. However, you can never divide both sides by zero, as division by zero is undefined and breaks the rules of algebra.

What happens if I only do something to one side?

If you change one side without changing the other, the statement is no longer true. The equals sign becomes a lie because the two expressions no longer hold the same value. You have permanently tipped the balance scale.

Do I have to do the same thing to both sides when I simplify an expression?

No. Simplifying one side of an equation—like combining like terms or distributing a number—does not change the actual value of that side. Because the value hasn't changed, the equation remains balanced, and you don't need to adjust the other side.

Why can you do the same thing to both sides of an equation?

An equation is simply a mathematical statement that two expressions have the exact same value. The equals sign acts as an anchor holding these two sides in perfect balance. Because both sides already represent the identical quantity, applying the exact same mathematical operation to both sides guarantees they will continue to be equal.

When we solve an algebraic equation, we are looking for the unknown value that makes the statement true. Doing the same thing to both sides allows us to change how the equation looks without changing the truth of the statement, letting us isolate the unknown variable step by step.

The balance scale analogy

Think of an equation like a perfectly balanced scale. If you have an unknown weight on the left side and a known weight on the right, the scale stays perfectly level because they equal each other. If you add a 5-pound weight to the left side, the scale will immediately tip. To restore the balance, you have to add exactly 5 pounds to the right side as well. What you do to one side, you must do to the other to keep the scale level.

The Properties of Equality

In mathematics, this intuitive concept is defined by the Properties of Equality. For example, the Addition Property of Equality states that if $a = b$ , then $a + c = b + c$ . There are similar properties for subtraction, multiplication, and division. These rules are the foundational laws that give us permission to manipulate algebraic equations confidently, knowing we aren't changing the final answer.

Why we do it: Isolating the variable

We use this rule to strip away the numbers surrounding our variable. If an equation has a $+ 4$ attached to a variable, we can use the Subtraction Property of Equality to subtract $4$ from both sides. This cancels out the $+ 4$ on the variable's side, bringing us one step closer to getting the variable completely by itself. We carefully choose inverse operations to undo whatever is happening to our variable.

Where students slip up

The most common mistake happens during multiplication or division. Students sometimes forget to apply the operation to the entire side of the equation. If your left side is $x + 3$ and you need to multiply both sides by $2$ , you must multiply the whole expression by $2$ , resulting in $2(x + 3)$ or $2x + 6$ . Multiplying only the $x$ will break the equality and tip the scale.

Worked through

Solve the equation $3x - 7 = 8$ by doing the same thing to both sides.

First, we want to undo the subtraction of 7. We do this by adding 7 to both sides of the equation.

$3x - 7 + 7 = 8 + 7$

$3x = 15$

Next, we need to undo the multiplication by 3. We do this by dividing both sides by 3.

$\frac{3x}{3} = \frac{15}{3}$

$x = 5$

By performing the same inverse operations on both sides, we kept the equation balanced and found that $x = 5$ .

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Where this comes from: OpenStax Prealgebra 2e, Chapter 2: Solving Linear Equations · Khan Academy, Algebra 1: Solving equations & inequalities