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 $x$.

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:

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

$= 2x + 6$