Concept Notes 1: The Concept of an Equation: Variables, Equality & The Two-Pan Balance Scale
Algebra is the branch of mathematics that uses letters and symbols to represent numbers and generalize arithmetic relationships. In Grade 7, we transition from expressions to Equations.
1. Variables, Constants & Algebraic Expressions
- Variable: A symbol (usually a letter such as $x, y, z, t$) that can take on different numerical values. A variable represents an unknown quantity.
- Constant: A quantity that has a fixed numerical value (e.g., $5, -12, \frac{3}{4}$).
- Algebraic Expression: A combination of variables and constants connected by arithmetic operations ($+, -, \times, \div$).
- Example: $4x + 7$, $3y - 5$, $\frac{z}{2} + 9$.
- An expression has no equality sign ($=$) and its value depends on the value assigned to the variable.
2. What is an Equation?
An Equation is a mathematical statement asserting that two algebraic expressions (or an expression and a constant) are equal in value.
- An equation must always contain an equals sign ($=$).
- Structure:
$$\underbrace{\text{Left Hand Side (LHS)}}_{\text{Expression 1}} = \underbrace{\text{Right Hand Side (RHS)}}_{\text{Expression 2}}$$
- At least one of the two sides must involve a variable.
| Expression (Phrase) | Equation (Statement of Equality) |
|---|---|
| $3x + 5$ | $3x + 5 = 26$ |
| $2y - 9$ | $2y - 9 = 15$ |
| $\frac{m}{4}$ | $\frac{m}{4} = 7$ |
| No '$=$' sign; value varies with variable | Has '$=$' sign; holds true only for specific value(s) |
3. The Two-Pan Balance Model (Equilibrium)
An equation is identical to a physical two-pan balance scale:
- The LHS represents the weights placed in the left pan.
- The RHS represents the weights placed in the right pan.
- The Equals Sign ($=$) signifies that the scale is in exact horizontal equilibrium.
The Four Invariance Properties of Balance:
- Addition Invariance: Adding the same number to both sides preserves balance:
$$\text{If } A = B, \text{ then } A + c = B + c$$
- Subtraction Invariance: Subtracting the same number from both sides preserves balance:
$$\text{If } A = B, \text{ then } A - c = B - c$$
- Multiplication Invariance: Multiplying both sides by the same non-zero number preserves balance:
$$\text{If } A = B, \text{ then } A \times c = B \times c \quad (c \ne 0)$$
- Division Invariance: Dividing both sides by the same non-zero number preserves balance:
$$\text{If } A = B, \text{ then } \frac{A}{c} = \frac{B}{c} \quad (c \ne 0)$$
4. What is a Solution (Root) of an Equation?
The specific numerical value of the variable that makes the LHS equal to the RHS is called the Solution or Root of the equation. When this value is substituted for the variable, the equation is said to be satisfied.
Verification by Substitution:
Consider the equation:
$$4x - 3 = 17$$
- If we test $x = 4$:
$$\text{LHS} = 4(4) - 3 = 16 - 3 = 13 \ne 17 \quad (\text{Not satisfied})$$
- If we test $x = 5$:
$$\text{LHS} = 4(5) - 3 = 20 - 3 = 17 = \text{RHS} \quad (\text{Satisfied!})$$
Therefore, $x = 5$ is the unique solution of the equation.