An algebraic expression like $3x + 5$ is a phrase (it has no equal sign). An equation is a complete mathematical sentence stating that two expressions have the exact same numerical value:
$$\mathbf{3x + 5 = 20}$$
The left-hand side ($\text{LHS}$) must balance the right-hand side ($\text{RHS}$). A solution (or root) is the specific number that makes the equation true when substituted for the variable.
An equation remains balanced if and only if you treat both sides equally:
- Add the same number to both sides: $x - 3 = 7 \implies x - 3 + 3 = 7 + 3 \implies x = 10$.
- Subtract the same number from both sides: $x + 4 = 9 \implies x + 4 - 4 = 9 - 4 \implies x = 5$.
- Multiply both sides by the same non-zero number: $\frac{x}{3} = 5 \implies \frac{x}{3} \times 3 = 5 \times 3 \implies x = 15$.
- Divide both sides by the same non-zero number: $4x = 24 \implies \frac{4x}{4} = \frac{24}{4} \implies x = 6$.
Example: Solve the equation showing each balancing step: $$2x + 7 = 19$$
Step 1 (Isolate term with variable): Subtract $7$ from both sides:
$$2x + 7 - 7 = 19 - 7 \implies 2x = 12$$
Step 2 (Isolate the variable): Divide both sides by $2$:
$$\frac{2x}{2} = \frac{12}{2} \implies \mathbf{x = 6}$$
Check: $\text{LHS} = 2(6) + 7 = 12 + 7 = 19 = \text{RHS}$. Verified!
Subtracting $7$ from the left side while leaving the right side as $19$ tilts the balance immediately! Whatever operation you perform on one side, you MUST perform identically on the other side.
Chemical equations in science (like $2\text{H}_2 + \text{O}_2 = 2\text{H}_2\text{O}$) obey this exact balance law: the number of atoms on the left must match the right.