Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Simplify algebraic expressions by collecting like terms.
• Solve linear equations like ax + b = c.
• Solve linear equations with the variable on both sides.
• Expand simple algebraic expressions.
• Solve simple word problems using equations.

Lesson summary

Algebraic expressions and equations are fundamental tools in mathematics. This week, we'll focus on simplifying algebraic expressions and solving simple equations. These skills are crucial not only for future mathematics courses but also for everyday problem-solving. Imagine trying to calculate the total cost of groceries when some prices are discounted, or figuring out how much data you have left on your cellphone plan - these scenarios often require algebraic thinking. In South Africa, budgeting and managing finances effectively are important skills, and understanding algebra can significantly aid in these areas. This week lays the foundation for more complex algebra in later grades.

Lesson notes

2.1 Algebraic Expressions: An algebraic expression is a combination of variables, constants, and operations (addition, subtraction, multiplication, division, exponentiation).

Variable: A letter that represents an unknown value (e.g., x, y, a, b).

Constant: A number that has a fixed value (e.g., 2, -5, 0.75).

Coefficient: The number multiplied by a variable (e.g., in 3x, 3 is the coefficient).

Like Terms: Terms that have the same variable(s) raised to the same power (e.g., 3x and 5x are like terms; 2y² and -7y² are like terms; 4x and 4x² are NOT like terms).

Simplifying Algebraic Expressions: Simplifying involves combining like terms. Only like terms can be added or subtracted.

Example 1: Simplify 3x + 5y - x + 2y.

Identify like terms: 3x and -x are like terms; 5y and 2y are like terms.

Combine like terms: (3x - x) + (5y + 2y) = 2x + 7y Simplified expression: 2x + 7y Example 2: Simplify 7a - 2b + 4a + b -

3. Identify like terms: 7a and 4a are like terms; -2b and b are like terms; -3 is a constant term.

Combine like terms: (7a + 4a) + (-2b + b) - 3 = 11a - b - 3 Simplified expression: 11a - b - 3 2.2 Algebraic Equations: An algebraic equation is a statement that two expressions are equal. It contains an equals sign (=). Solving an equation means finding the value(s) of the variable(s) that make the equation true. Solving Equations of the form ax + b = c: To solve an equation, we isolate the variable on one side of the equation. We use inverse operations to "undo" the operations performed on the variable. Remember, whatever you do to one side of the equation, you must do to the other side to maintain equality.

Example 3: Solve 2x + 3 =

7. Subtract 3 from both sides: 2x + 3 - 3 = 7 - 3 Simplify: 2x = 4 Divide both sides by 2: 2x / 2 = 4 / 2 Simplify: x = 2 Therefore, the solution is x =

2. Solving Equations with the variable on both sides: Example 4: Solve 5x - 2 = 3x +

4. Subtract 3x from both sides: 5x - 2 - 3x = 3x + 4 - 3x Simplify: 2x - 2 = 4 Add 2 to both sides: 2x - 2 + 2 = 4 + 2 Simplify: 2x = 6 Divide both sides by 2: 2x / 2 = 6 / 2 Simplify: x = 3 Therefore, the solution is x = 3. 2.3 Distributive Property: The distributive property states that a(b + c) = ab + ac. This means you multiply the term outside the parentheses by each term inside the parentheses.

Example 5: Expand 3(x + 2).

Multiply 3 by x: 3 x = 3x Multiply 3 by 2: 3 2 = 6 Result: 3x + 6 Example 6: Expand -2(y - 4).

Multiply -2 by y: -2 y = -2y Multiply -2 by -4: -2 -4 = 8 (Remember a negative times a negative is a positive)

Result: -2y + 8 Guided Practice (With Solutions)

Question 1: Simplify the expression: 4a + 7b - 2a + b - 3 Solution: Identify like terms: 4a and -2a are like terms; 7b and b are like terms.

Group like terms: (4a - 2a) + (7b + b) - 3 Combine like terms: 2a + 8b - 3 Final Answer: 2a + 8b - 3

Commentary: We grouped the like terms together before combining them to avoid errors. The constant term -3 cannot be combined with any other terms.

Question 2: Solve the equation: 3x - 5 = 10 Solution: Add 5 to both sides: 3x - 5 + 5 = 10 + 5 Simplify: 3x = 15 Divide both sides by 3: 3x / 3 = 15 / 3 Simplify: x = 5 Final Answer: x = 5

Commentary: We used inverse operations (addition and division) to isolate the variable 'x'.

Question 3: Solve the equation: 2y + 1 = y - 4 Solution: Subtract y from both sides: 2y + 1 - y = y - 4 - y Simplify: y + 1 = -4 Subtract 1 from both sides: y + 1 - 1 = -4 - 1 Simplify: y = -5 Final Answer: y = -5

Commentary: By subtracting 'y' from both sides, we managed to get all the 'y' terms on one side of the equation.

Question 4: Expand the expression: 5(2x - 3)

Solution: Multiply 5 by 2x: 5 2x = 10x Multiply 5 by -3: 5 -3 = -15 Final Answer: 10x - 15

Commentary: Remember to multiply the term outside the parentheses by each term inside. Independent Practice (Questions Only)

Simplify: 6p - 2q + 4q - p + 7 Solve: 4x + 2 = 14 Solve: 7y - 3 = 5y + 1 Expand: 2(3a + 4)

Expand: -3(2b - 5)

Solve: x/2 + 3 = 7 Simplify: 5m + 3n - 2m - 7 + n Solve: 6x - 4 = 2x + 8 Expand and Simplify: 4(y + 2) - 3y Solve the word problem: A cellphone costs R50 more than a pair of shoes. If the cellphone costs R250, what is the cost of the pair of shoes? (Let the cost of the shoes be x)