Scaling the Scale
Just as we can add to both sides, we can multiply or divide both sides by the same (non-zero) number. This is the Multiplication Property of Equality.
Inverses Revisited
- To undo multiplication, divide.
- To undo division, multiply.
Worked Examples
Example 1: Undoing Multiplication
Solve: \(4x = 24\)
- The variable is multiplied by 4.
- Divide both sides by 4.
- \(\frac{4x}{4} = \frac{24}{4}\)
- Result: \(x = 6\)
Example 2: Undoing Division
Solve: \(\frac{x}{5} = 10\)
- The variable is divided by 5.
- Multiply both sides by 5.
- \(5 \times \frac{x}{5} = 5 \times 10\)
- Result: \(x = 50\)
Example 3: Negative Coefficients
Solve: \(-3x = 15\)
- Divide by -3.
- \(\frac{-3x}{-3} = \frac{15}{-3}\)
- Result: \(x = -5\)
Example 4: Fraction Coefficients
Solve: \(\frac{2}{3}x = 8\)
- To undo a fraction, multiply by its Reciprocal (Lesson 7).
- \(\frac{3}{2} \times \frac{2}{3}x = \frac{3}{2} \times 8\)
- \(x = \frac{24}{2} = 12\)
- Result: \(x = 12\)
The Bridge to Quantum Mechanics
In the Schrödinger Equation, the energy \(E\) is often "scaled" by constants like \(\hbar\) or \(m\). To find the actual value of the energy, we have to "move" these constants to the other side of the equals sign using multiplication and division. If you can't confidently scale an equation, you'll never be able to extract the physical energy levels of an atom from its mathematical form.