An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation signs. For example, $3x + 5$ is an expression where $x$ is a variable.

Key Terms

• Variable: A letter that represents an unknown number (e.g., $x$, $y$, $z$)
• Constant: A fixed number on its own (e.g., 5, -3, 7)
• Coefficient: A number multiplied by a variable (e.g., in $4x$, the coefficient is 4)
• Term: A single part of an expression (e.g., $3x$ and $5$ are terms in $3x + 5$)

Simplifying Expressions

We simplify expressions by combining like terms—terms that have the same variable raised to the same power.

Example: Simplify $3x + 5 + 2x - 2$

Solution: $3x + 5 + 2x - 2 = (3x + 2x) + (5 - 2) = 5x + 3$

Order of Operations (BODMAS)

When evaluating expressions, follow BODMAS:

1. Brackets
2. Orders (powers and roots)
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Example: Simplify $12 \div 3 + 5 \times 2$

Solution: $12 \div 3 + 5 \times 2 = 4 + 10 = 14$

An equation is a statement that two expressions are equal, using the sign $=$. Solving an equation means finding the value of the variable that makes the equation true.

Simple Linear Equations

A linear equation has the highest power of the variable as 1 (e.g., $2x + 3 = 11$).

Example: Solve $2x + 3 = 11$

Solution: $2x + 3 = 11$ $2x = 11 - 3$ $2x = 8$ $x = \frac{8}{2}$ $x = 4$

Checking Your Answer

Always substitute your answer back into the original equation: $2(4) + 3 = 8 + 3 = 11 \checkmark$

Simultaneous equations are two or more equations with the same unknowns. We solve them to find values that satisfy both equations at the same time.

Methods of Solving

1. Substitution Method

Example: Solve: $x + y = 10$ $x - y = 4$

Solution: From the first equation: $x = 10 - y$

Substitute into the second equation: $(10 - y) - y = 4$ $10 - 2y = 4$ $-2y = 4 - 10$ $-2y = -6$ $y = 3$

Now find $x$: $x = 10 - y = 10 - 3 = 7$

Answer: $x = 7$, $y = 3$

2. Elimination Method

Example: Solve: $2x + y = 8$ $x + y = 5$

Solution: Subtract the second equation from the first: $(2x + y) - (x + y) = 8 - 5$ $x = 3$

Substitute $x = 3$ into $x + y = 5$: $3 + y = 5$ $y = 2$

Answer: $x = 3$, $y = 2$

An inequality shows that two expressions are not equal. The symbols are:

• $<$ means "less than"
• $>$ means "greater than"
• $\leq$ means "less than or equal to"
• $\geq$ means "greater than or equal to"

Solving Inequalities

Inequalities are solved similarly to equations, with one important rule:

When multiplying or dividing both sides by a negative number, the inequality sign reverses direction.

Example 1: Solve $x + 3 > 7$

Solution: $x + 3 > 7$ $x > 7 - 3$ $x > 4$

Example 2: Solve $3x \leq 12$

Solution: $3x \leq 12$ $x \leq \frac{12}{3}$ $x \leq 4$

Example 3: Solve $-2x > 6$

Solution: $-2x > 6$ $x < \frac{6}{-2}$ (sign reverses) $x < -3$

Representing Inequalities on a Number Line

Inequalities can be shown graphically:

• $x > 4$ is shown as an open circle at 4 with shading to the right
• $x \geq 4$ is shown as a closed (filled) circle at 4 with shading to the right
• $x < -3$ is shown as an open circle at -3 with shading to the left

In Tanzania, algebra is used in everyday situations such as calculating profit and loss in small businesses. For example, if a vendor at a market in Dar es Salaam buys $x$ mangoes at TZS 500 each and sells them for a total of TZS 15,000, they can use a simple equation ($500x = 15,000$) to find that they sold 30 mangoes. Similarly, shopkeepers use inequalities to set minimum prices or determine bulk discounts for their customers.