Learning Resources

Algebraic Expressions

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 07 Pathway: N/A

First Principles

Objective: Form, simplify, and evaluate algebraic expressions by identifying and collecting like terms.

An algebraic expression is a mathematical statement that combines numbers, operation symbols (like \(+\) and \(-\)), and letters called variables. Variables represent unknown or changing values. In an expression such as \(4x + 7\):

\(x\) is the variable (an unknown quantity).
\(4\) is the coefficient (the multiplier of the variable).
\(7\) is a constant (a fixed numerical value with no variable attached).

Concrete East African Market Scenario: Imagine visiting Gikomba or Karatina market. If a trader sells mangoes at KSh \(m\) each and pineapples at KSh \(p\) each, buying 3 mangoes and 2 pineapples costs \(3m + 2p\). The terms \(3m\) and \(2p\) represent two entirely different fruits, so they cannot be combined into a single term like \(5mp\). They must stay separate: \(3m + 2p\).

The Principle of Like Terms: Like terms are terms that contain the exact same variable(s) raised to the exact same power. For example:

\(5a\) and \(3a\) are like terms: \(5a + 3a = 8a\).
\(4x^2\) and \(7x^2\) are like terms: \(4x^2 - 7x^2 = -3x^2\).
\(3x\) and \(3y\) are unlike terms (different variables).
\(2x\) and \(2x^2\) are unlike terms (different powers).

Core Rule: To simplify expressions, collect like terms by adding or subtracting their coefficients while keeping the variable factor unchanged: \[k_1 x + k_2 x = (k_1 + k_2)x\]

x terms

y terms

Constants

Simplified Result:

Sort terms above

Key Formulas

Key algebraic rules and reference formulas for Grade 7 CBE:

\[\text{Structure of an Algebraic Term: } ax^n\]

Where \(a\) is the numerical coefficient, \(x\) is the base variable, and \(n\) is the index (power). When no index is written, \(n = 1\). When no coefficient is written, \(a = 1\).

\[\text{Distributive Property: } a(b + c) = ab + ac \quad \text{and} \quad a(b - c) = ab - ac\]

Multiply the factor outside the brackets by every term inside the brackets before collecting like terms.

\[\text{Collecting Like Terms: } k_1 x + k_2 x = (k_1 + k_2)x\]

Combine coefficients of identical variable terms while keeping the variable part constant.

\[\text{Evaluation by Substitution: } f(x) = ax + b \implies f(k) = a(k) + b\]

To evaluate an algebraic expression, replace each variable with its assigned numerical value using brackets, then calculate following BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

\[\text{Sign Rules for Negative Numbers:}\]

\((-) \times (+) = (-)\)
\((-) \times (-) = (+)\)
\((-a)^2 = (-a) \times (-a) = +a^2\)

Worked Examples

Example 1 (Easy — 1-Step Substitution):
Evaluate the expression \(2m + 5\) when \(m = 6\).

Step-by-step Solution:

Identify the variable to replace: \(m = 6\).
Substitute \(6\) into the expression in place of \(m\): \[2(6) + 5\]
Perform multiplication before addition (BODMAS): \[12 + 5 = 17\]

Answer: \(17\)

Example 2 (Medium — Collecting Like Terms & Simplifying):
Simplify the expression \((6m + 17) - (6m + 14)\) and find its numerical value.

Step-by-step Solution:

Distribute the negative sign across the second bracket: \((6m + 17) - (6m + 14) = 6m + 17 - 6m - 14\)
Group the \(m\)-terms together and the constant terms together: \[(6m - 6m) + (17 - 14)\]
Simplify each group: \[0m + 3 = 3\]

Answer: \(3\)

Example 3 (Hard — Multi-Step Expansion & Quadratic Evaluation):
A civil engineer models the load capacity of a bridge beam using the formula \(L = 2d^2 - 5d + 7\), where \(d\) is the depth in metres. Calculate the load capacity when \(d = 6\text{ m}\).

Step-by-step Solution:

Substitute \(d = 6\) into the quadratic expression: \[L = 2(6)^2 - 5(6) + 7\]
Evaluate the exponent (Order/Power): \[(6)^2 = 36\]
Perform multiplications: \[2(36) - 5(6) + 7 = 72 - 30 + 7\]
Perform subtraction and addition from left to right: \[72 - 30 = 42\] \[42 + 7 = 49\]

Answer: \(49\text{ kN}\)

Common Mistakes

Misconception 1 Adding unlike terms into a single combined term, e.g., writing \(3x + 2y = 5xy\).

Why it feels right In everyday speech, we combine items together (e.g., 3 pens and 2 pencils = 5 items). Learners mistakenly add the numbers and glue the letters together.

The Mathematical Reality \(x\) and \(y\) represent different unknown quantities. \(3x + 2y\) means \(3 \times x + 2 \times y\), whereas \(5xy\) means \(5 \times x \times y\). You cannot combine terms with different variable bases. \(3x + 2y\) is already in its simplest form.

Misconception 2 Incomplete bracket expansion, e.g., writing \(3(x + 4) = 3x + 4\) or \(-(x + 5) = -x + 5\).

Why it feels right The eye focuses on multiplying the first term \(3 \times x\) and forgets that the multiplier outside applies to all items inside the brackets.

The Mathematical Reality The distributive property requires multiplying every item: \(3(x + 4) = 3(x) + 3(4) = 3x + 12\). Similarly, \(-(a + b) = -1(a + b) = -a - b\).

Misconception 3 Confusing squaring a negative variable value, e.g., calculating \((-2)^2 = -4\).

Why it feels right The learner sees a negative sign and carries it through to the final answer.

The Mathematical Reality When \(m = -2\), \(m^2 = (-2) \times (-2) = +4\). A negative number multiplied by another negative number always produces a positive product.

Real World

Boda-Boda & Matatu Fare Calculations: A boda-boda rider in Kisumu charges a base flag drop of KSh 50 plus KSh 30 per kilometre \(k\). The fare is represented by the linear expression \(50 + 30k\). If a passenger travels 4 km, the fare is \(50 + 30(4) = \text{KSh } 170\).

Agricultural Shamba Management: A tea farmer in Kericho plants seedlings in square plots of side length \(m\) metres. Preparing the soil costs KSh 2 per square metre (Area \(= m^2\)) and fencing the boundary costs KSh 3 per metre (Perimeter \(= 4m\)). The total investment expression is \(2m^2 + 3(4m) = 2m^2 + 12m\).

M-Pesa / Mobile Money Fees: Sending money often includes a fixed network levy plus a percentage rate: \(\text{Fee} = 0.01x + 15\), where \(x\) is the principal amount transferred.

School Supplies Budgeting: A headteacher ordering \(b\) exercise books at KSh 45 and \(p\) biros at KSh 15 calculates the total budget as \(45b + 15p\).

Practice

Baraka drives a boda-boda. He charges KES 2m for each kilometre travelled plus a fixed waiting fee of KES 5. If the distance travelled is m = 6 km, how much does a passenger pay in total in Kenyan Shillings? (Type only the number, e.g., 42)

Review the concepts above.

Kiprono drives a school bus. The bus uses 3x litres of diesel for the morning route and 2y litres for the afternoon route. If x = 2 and y = 5, how many litres of diesel are used in total? (Type only the number, e.g., 42)

Review the concepts above.

Gathoni runs a small hair salon. On Monday she earned (6m + 17) Kenyan shillings. On Tuesday she spent (6m + 14) shillings on hair products. What is her net profit for the two days (Monday earnings minus Tuesday expenses)? (Type only the number, e.g., 42)

Review the concepts above.

Eunice sells potatoes at the market. Her profit in Kenyan Shillings is modelled by the formula 3m^2 + 2, where m is the number of extra bundles sold. One day she mistakenly records m = -2 instead of m = 2. What profit value in KSh does her calculation give? (Type only the number, e.g., 42)

Review the concepts above.

A civil engineer uses the formula 2d^2 - 5d + 7 to estimate the load capacity (in kN) of a bridge beam, where d is the depth in metres. What is the load capacity when the depth d = 6 m? (Type only the number, e.g., 42)

Review the concepts above.

A farmer is preparing a square garden with side length m metres. Preparing the soil costs KES 2 per square metre and installing wire-mesh fencing around the perimeter costs KES 3 per metre. The total cost is modelled by 2m^2 + 12m. If the side length is m = 5 metres, what is the total cost in Kenyan Shillings? (Type only the number, e.g., 42)

Review the concepts above.