Equations and Equivalent Transformations¶
What Is an Equation?¶
An equation is a statement of the form \(A = B\), where \(A\) and \(B\) are mathematical expressions. The equals sign "\(=\)" means: both sides represent the same value.
Examples.
- \(3 + 4 = 7\) — a true statement.
- \(2x + 1 = 9\) — a statement whose truth value depends on \(x\). For \(x = 4\): true. For \(x = 3\): false.
Equivalence Transformations¶
An equivalence transformation modifies an equation while preserving its solution set. The new equation has exactly the same solutions as the original.
Permitted Operations¶
| Operation | Formula | Condition |
|---|---|---|
| Add to both sides | \(A = B \iff A + c = B + c\) | — |
| Subtract from both sides | \(A = B \iff A - c = B - c\) | — |
| Multiply both sides | \(A = B \iff A \cdot c = B \cdot c\) | \(c \neq 0\) |
| Divide both sides | \(A = B \iff \frac{A}{c} = \frac{B}{c}\) | \(c \neq 0\) |
The principle: the same operation is applied to both sides. The equation remains balanced.
Example: Solving \(2x + 3 = 11\)¶
| Step | Equation | Operation |
|---|---|---|
| 1 | \(2x + 3 = 11\) | Starting equation |
| 2 | \(2x = 8\) | \(-3\) on both sides |
| 3 | \(x = 4\) | \(\div 2\) on both sides |
Every line has the same solution set: \(\{4\}\).
Non-Equivalence Transformations¶
Not every operation preserves the solution set:
- Multiplication by 0: From \(x = 3\) one obtains \(0 = 0\) — every \(x\) is now a "solution". Solution set enlarged.
- Squaring: From \(x = -2\) one obtains \(x^2 = 4\), which also admits \(x = 2\). Solution set enlarged.
- Division by an expression containing the variable: From \(x^2 = 2x\), dividing by \(x\) yields \(x = 2\). The solution \(x = 0\) is lost.
Equations with Fractions¶
For equations involving fractions, multiply both sides by the least common denominator:
Example. \(\frac{x}{3} + \frac{1}{2} = \frac{5}{6}\)
Multiply by \(6\) (lcm of \(3, 2, 6\)):
Systems of Equations¶
A system of equations consists of several equations with several unknowns. The solution must satisfy all equations simultaneously.
Example. \(\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}\)
Adding both equations: \(2x = 6\), so \(x = 3\). Substituting: \(y = 2\).
Summary¶
| Concept | Meaning |
|---|---|
| Equation | Statement \(A = B\) |
| Equivalence transformation | Operation preserving the solution set |
| Permitted | \(+c\), \(-c\), \(\cdot c\) (\(c \neq 0\)), \(\div c\) (\(c \neq 0\)) on both sides |
| Not permitted | \(\cdot 0\), squaring, division by variable (without case distinction) |
References¶
- Courant, Richard; Robbins, Herbert: What Is Mathematics? Oxford University Press, 2nd edition, 1996.