Equations – The Language of Algebra
An equation is the mathematical equivalent of a balanced scale. Whatever you do to one side, you must do exactly the same to the other. This principle is the foundation of all equation solving.
The equals sign itself was invented by the Welsh mathematician Robert Recorde in his 1557 book The Whetstone of Witte, who chose two parallel lines specifically “bicause noe 2 thynges can be moare equalle” — his own explanation for why two identical horizontal lines were the perfect symbol for equality. Before Recorde, mathematicians had to write out phrases like “is equal to” in full, which made even short equations cumbersome to read and copy.
What Is an Equation?
An equation is a mathematical statement that says two expressions are equal. It always contains an equals sign (=). The goal of solving an equation is to find the value of the unknown variable that makes the statement true.
Equations vs Expressions
| Expression | Equation | |
|---|---|---|
| Contains = | No | Yes |
| Example | 3x + 2 | 3x + 2 = 11 |
| Can solve for x? | No | Yes |
| Has a solution | No | Yes (one or more values) |
Types of Equations
| Type | Example | Highest Power |
|---|---|---|
| Linear | 2x + 3 = 9 | 1 |
| Quadratic | x squared - 5x + 6 = 0 | 2 |
| Cubic | x cubed = 8 | 3 |
| Simultaneous | x + y = 5 and 2x - y = 1 | Two unknowns |
The Balance Principle
Think of an equation as a balanced scale, with the equals sign as the pivot. Both sides currently weigh the same. To keep it balanced, any operation you perform on one side, you must also perform on the other side — add the same amount to both pans, or remove the same amount from both pans, and the scale stays level.
Removing the same amount (3) from both pans keeps the scale level — it doesn't matter that the pans hold different-looking things, only that both sides lose the same amount. That's the entire balance principle in one picture: whatever happens to one pan must happen to the other, or the scale (and the equation) stops being true.
The Golden Rule – You Are Never “Moving” a Number
Many students are taught to “move a term to the other side and flip its sign or operation” (+ becomes −, − becomes +, × becomes ÷, ÷ becomes ×). This shortcut gives the right answer, but it's easy to apply incorrectly if you don't understand why it works — which is exactly where mistakes like dropping a minus sign creep in.
Here is what is really happening: nothing ever “moves.” Instead, you choose the opposite (inverse) operation of whatever is being done to the term you want to remove, and you apply that opposite operation to both sides of the equation, in full, at the same time. On the side where the term started, the operation cancels itself out and the term disappears. On the other side, that same operation is now genuinely being carried out on the numbers already there — which is why a new term appears there, already correctly signed.
The Four Operation Pairs
To undo an operation, always apply its opposite to both sides:
| Term is being… | Apply this to both sides | It looks like… |
|---|---|---|
| Added ( + a ) | Subtract a from both sides | “+a becomes −a” |
| Subtracted ( − a ) | Add a to both sides | “−a becomes +a” |
| Multiplied ( × a ) | Divide both sides by a | “×a becomes ÷a” |
| Divided ( ÷ a ) | Multiply both sides by a | “÷a becomes ×a” |
The right-hand column is just what it looks like happened, once you simplify. The middle column is what you must actually do, on both sides, every time.
Undoing Addition and Subtraction
The 5 is added to x. To remove it, subtract 5 from both sides — the whole equation, not just the left side:
x + 5 − 5 = 12 − 5
On the left, +5 and −5 cancel out, leaving just x. On the right, 12 − 5 is a real subtraction that gives 7. Therefore x = 7.
This time the 5 is subtracted from x. The opposite of subtracting 5 is adding 5, so add 5 to both sides:
x − 5 + 5 = 12 + 5
On the left, −5 and +5 cancel out, leaving x. On the right, 12 + 5 = 17. Therefore x = 17. Notice that because the 5 was being subtracted, we added on both sides — the opposite operation, not the same one.
Undoing Multiplication and Division
Here y is being multiplied by 3. The opposite of multiplying by 3 is dividing by 3, so divide both sides by 3:
3y ÷ 3 = 18 ÷ 3
On the left, multiplying by 3 and dividing by 3 cancel out, leaving y. On the right, 18 ÷ 3 = 6. Therefore y = 6.
Here x is being divided by 4. The opposite of dividing by 4 is multiplying by 4, so multiply both sides by 4:
(x ÷ 4) × 4 = 9 × 4
On the left, dividing by 4 and multiplying by 4 cancel out, leaving x. On the right, 9 × 4 = 36. Therefore x = 36.
The Trickiest Case – A Negative Coefficient
Equations like 25 − 3x = 453, where a number comes first and the x-term is being subtracted, cause the most confusion. Students often aren't sure how to deal with the leading 25, and then forget that −3x needs to stay negative when they isolate x. Working through it in two clear, separate steps — and always applying each operation to both sides — removes the guesswork.
Step 1 – deal with the 20 first. On the left side, 20 is being added (the term is “+20”, even though it's written first). To remove it, subtract 20 from both sides — the whole equation:
20 − 3x − 20 = 5 − 20
On the left, +20 and −20 cancel, leaving −3x untouched — the minus sign in front of the 3x was never affected, because we only ever added or subtracted the 20. On the right, 5 − 20 = −15. So now:
−3x = −15
Step 2 – deal with the −3. Now x is being multiplied by −3 (not +3 — the negative sign is part of the coefficient). The opposite of multiplying by −3 is dividing by −3, so divide both sides by −3:
(−3x) ÷ (−3) = (−15) ÷ (−3)
On the left, dividing by −3 cancels the ×(−3), leaving x. On the right, a negative divided by a negative is positive: −15 ÷ −3 = 5. Therefore x = 5.
Check: 20 − 3(5) = 20 − 15 = 5. Correct!
Step 1 – remove the 25. The 25 is added on the left, so subtract 25 from both sides:
25 − 3x − 25 = 453 − 25
Left side: +25 and −25 cancel, leaving −3x exactly as it was — still negative. Right side: 453 − 25 = 428. So now:
−3x = 428
Step 2 – remove the −3. x is being multiplied by −3, so divide both sides by −3 (the full coefficient, sign included):
(−3x) ÷ (−3) = 428 ÷ (−3)
Left side: cancels to x. Right side: a positive divided by a negative is negative: 428 ÷ (−3) = −142.67 (to 2 decimal places), or exactly −428⁄3. Therefore x = −428⁄3 ≈ −142.67.
This example doesn't give a whole number — and that's fine. The method is identical whether the answer is a tidy integer or a fraction. The mistake to avoid is writing x = 428 ÷ 3 (dropping the negative sign) instead of x = 428 ÷ (−3). The negative sign belongs to the 3, not to a separate “moving” step, so it must be included in the division.
Two-Step Equations
Longer equations just chain the same idea together: undo addition/subtraction first, then undo multiplication/division.
Subtract 7 from both sides: 4x + 7 − 7 = 31 − 7, which gives 4x = 24.
Divide both sides by 4: 4x ÷ 4 = 24 ÷ 4, which gives x = 6.
Subtract 9 from both sides: −2x + 9 − 9 = 1 − 9, which gives −2x = −8.
Divide both sides by −2: (−2x) ÷ (−2) = (−8) ÷ (−2), which gives x = 4 (negative ÷ negative = positive).
Checking a Solution
Substitute the answer back into the original equation. Both sides must give the same value.
Left side: 3 times 6 = 18. Right side: 18. Both sides equal. Solution verified!
Key Takeaways
- An equation has an equals sign; an expression does not.
- Treat an equation like a balanced scale — whatever you do to one side, do to the other.
- Always check your solution by substituting it back into the original equation.