Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. This fundamental concept allows us to generalise mathematical ideas beyond simple arithmetic.

**Introduction to Algebraic Representation**

Algebra uses letters like $x$, $y$, $z$ to symbolize variables or unknown quantities, allowing for the formulation of equations and expressions that can universally apply beyond specific numerical instances.

- Variables: Symbols for unknown numbers.
- Constants: Known values.
- Coefficients: Numbers multiplying a variable.

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**Algebraic Operations**

**Example 1: Solving an Equation**

**Problem**: Find $x$ in $x - 3 = 4$.

**Solution**:

$x - 3 = 4$

$x = 4 + 3$

$x = 7$

**Example 2: Substitution**

**Problem**: Given $x = 5$, evaluate $2x + 3$.

**Solution**:

$2x + 3 = 2(5) + 3$

$= 10 + 3$

$= 13$

**Algebraic Expressions**

**Example 3: Simplifying Expressions**

**Problem**: Simplify $2x + 3x - 5$.

**Solution**:

**Representation in Equations**

**Example 4: Formulating Equations**

**Problem**: Represent "a number plus three equals eleven" as an equation.

**Solution**:

Let the number be $x$.

$x + 3 = 11$**Solving Algebraic Equations**

**Example 5: Solving for a Variable**

**Problem**: Solve $2x = 10$.

**Solution**:

$2x = 10$

$x = \frac{10}{2}$

$x = 5$

**Practice Problems**

**1. Solve for **$y$**: **$3y + 4 = 19$

**Solution:**

$3y + 4 = 19$

$3y = 15$

$y = \frac{15}{3}$

$y = 5$

**2. Given **$z = 2$**, find**: $4z - 1$.

**Solution:**

$4z - 1 = 4(2) - 1$

$= 8 - 1$

$= 7$

**3. Simplify**: $3(x + 2) - x$

**Solution:**

$3(x + 2) - x = 3x + 6 - x$

$= 2x + 6$