Why must I do the same operation to both sides of an equation?
Quick answer
An equation says its two sides have equal value. Doing the same valid operation to both sides keeps them equal, like changing both sides of a balanced scale in the same way. For example, subtracting 3 from both sides of x + 3 = 8 gives x = 5. Subtracting 3 from only one side changes the relationship, so that step does not preserve the equation.
The question, the details, the next step.
Why do I have to do the same thing to both sides? Is it just a rule to memorize?
It helps to think of an equation as saying, “the value on the left equals the value on the right.” If both sides start equal, adding or subtracting the same quantity from each keeps them equal. Multiplying both sides by the same number also preserves equality; dividing both sides preserves it when the divisor is not zero. [S1] [S2]
Imagine a balanced scale: changing both sides in the same way keeps the balance. The analogy is useful, but the key idea is the equality statement—not a physical scale. A solving step should preserve the equation’s truth so you can work toward isolating the unknown.
Can you show how that works when solving an equation?
Take x + 3 = 8. The + 3 is attached to x, so subtract 3 from both sides to undo it:
x + 3 - 3 = 8 - 3- Simplify:
x = 5
The equation stays balanced because each side changed by the same amount. Check the result by putting 5 in place of x: 5 + 3 = 8, which is true. Substitution is a useful way to check a proposed solution. [S1]
Another example is 2x = 10. Divide both sides by 2, giving x = 5; checking gives 2 × 5 = 10. Dividing by zero is not allowed, so division steps need a nonzero divisor. [S2]
What does an invalid step look like? I sometimes move a number across the equals sign.
“Move it across” is shorthand for doing an operation to both sides. For x + 3 = 8, subtracting 3 from just the right side would give x + 3 = 5. That is not the same equation: if x = 5, its sides would be 8 and 5, not equal.
Compare these steps:
- Valid:
x + 3 = 8becomesx + 3 - 3 = 8 - 3, thenx = 5. - Invalid:
x + 3 = 8becomesx + 3 = 5by changing only the right side.
A quick test is to plug the proposed answer into the original equation. If it does not make the original left and right sides equal, the answer—or a step leading to it—needs another look.
Is every same operation safe, and how can I keep track of my steps?
For the basic solving moves here, add or subtract the same quantity on both sides, multiply both sides by the same number, or divide both sides by the same nonzero number. [S1] [S2] Still, a step can preserve equality while making an equation less useful. For instance, multiplying both sides by zero turns any equation into 0 = 0, which loses information about the unknown. So choose an operation that helps isolate the variable, and avoid steps that discard information.
A copyable setup is:
Original equation: [left side] = [right side]
Operation on both sides: [operation]
New equation: [simplified left side] = [simplified right side]
Check: substitute [proposed value] into the original equation.
For example: x + 3 = 8 → subtract 3 from each side → x = 5 → check: 5 + 3 = 8. Writing the operation explicitly makes it easier to see whether both sides were treated alike.
A practical checklist
Write the equation before changing it.
Name the operation you are using, such as subtract 3 or divide by 2.
Apply that operation to both sides.
For division, make sure the divisor is not zero.
Substitute the proposed answer into the original equation and check both sides.
A few more useful answers
Can I add different expressions to the two sides?
A step that adds different amounts does not generally preserve equality. Use the same quantity on both sides unless you have separately shown why the transformation is valid.
Does checking the answer replace showing my solving steps?
No. A check can confirm that a proposed value works in the original equation, but clear steps show how you reached it and help reveal mistakes.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Addition and Subtraction Properties of Equality openstax.org · Checked
- [S2] OpenStax Elementary Algebra 2e: Multiplication and Division Properties of Equality openstax.org · Checked