Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Simplify expressions using all four basic operations.
• Evaluate expressions by substituting values.
• Solve linear equations with one variable.
• Create equations from word problems.
• Use the distributive property to expand expressions.

Lesson summary

Algebraic expressions and equations form the foundation of much of advanced mathematics. This week, we delve deeper into manipulating expressions and solving equations. This skill is crucial not only for further studies in mathematics, science, and engineering but also for everyday problem-solving. Imagine you're running a small business selling vetkoek at the local taxi rank – you'll need algebra to calculate your profits, costs, and selling prices! Understanding algebraic principles empowers you to make informed decisions in various aspects of life, from managing your personal finances to understanding scientific data presented in the news.

Lesson notes

2. 1. Algebraic Expressions An algebraic expression is a combination of variables (represented by letters, like x, y, a, b), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division, powers). For example, 3x + 5y – 2 is an algebraic expression.

Terms: Parts of an algebraic expression separated by '+' or '-' signs. In the expression 3x + 5y – 2, the terms are 3x, 5y, and -

2. Coefficients: The numerical factor of a term. In 3x, the coefficient is

3. In -5y, the coefficient is -

5. Constants: Terms without any variables. In 3x + 5y – 2, the constant is -

2. Like Terms: Terms that have the same variable raised to the same power. For example, 2x and 5x are like terms, but 2x and 2x² are not. We can only add or subtract like terms.

Example 1: Simplifying Algebraic Expressions Simplify the expression: 5a + 3b – 2a + 7b Step 1: Identify like terms. 5a and -2a are like terms. 3b and 7b are like terms.

Step 2: Group the like terms together: (5a – 2a) + (3b + 7b)* Step 3: Combine the like terms: 3a + 10b 2.

2. Algebraic Equations An algebraic equation is a statement that two algebraic expressions are equal. It contains an equals sign (=). For example, 2x + 3 = 7 is an equation.

Solving an Equation: Finding the value(s) of the variable(s) that make the equation true.

Inverse Operations: Operations that "undo" each other (e.g., addition and subtraction, multiplication and division). We use inverse operations to isolate the variable in an equation.

Maintaining Equality: Whatever operation you perform on one side of the equation, you must also perform on the other side to keep the equation balanced.

Example 2: Solving a Linear Equation Solve the equation: 4x – 5 = 11 Step 1: Isolate the term with the variable.

Add 5 to both sides of the equation: 4x – 5 + 5 = 11 + 5 4x = 16 Step 2: Isolate the variable.

Divide both sides of the equation by 4: 4x / 4 = 16 / 4 x = 4 Example 3: Equations with brackets Solve the equation: 2(x + 3) = 10 Step 1: Apply the distributive property to remove the brackets.

Multiply 2 by both x and 3: 2 x + 2 3 = 10 2x + 6 = 10 Step 2: Subtract 6 from both sides: 2x + 6 - 6 = 10 - 6 2x = 4 Step 3: Divide both sides by 2: 2x / 2 = 4 / 2 x = 2 2.

3. Distributive Property The distributive property states that a(b + c) = ab + ac. This means you multiply the term outside the brackets by each term inside the brackets.

Example 4: Applying the Distributive Property Expand the expression: 3(2x – 5)

Step 1: Multiply 3 by 2x: 3 2x = 6x Step 2: Multiply 3 by -5: 3 (-5) = -15 Step 3: Combine the results: 6x – 15 Example 5: A more complex expression Simplify: 4(x + 2) - 2(x - 1)

Step 1: Distribute the 4 into the first bracket: 4x + 8 Step 2: Distribute the -2 into the second bracket: -2x + 2 Step 3: Combine the two expanded expressions: 4x + 8 - 2x + 2 Step 4: Group like terms: (4x - 2x) + (8 + 2)* Step 5: Simplify: 2x + 10 Guided Practice (With Solutions)

Question 1: Simplify the expression: 7y – 4x + 2y + 6x – 3 Solution: Step 1: Identify like terms: 7y and 2y are like terms. -4x and 6x are like terms. -3 is a constant term.

Step 2: Group the like terms together: (7y + 2y) + (-4x + 6x) – 3 Step 3: Combine the like terms: 9y + 2x – 3

Commentary: Be careful to include the signs (+ or -) in front of each term when grouping.

Question 2: Evaluate the expression 2a² – 3b + 5 when a = 3 and b = -

2. Solution: Step 1: Substitute the given values into the expression: 2(3)² – 3(-2) + 5 Step 2: Simplify the expression following the order of operations (PEMDAS/BODMAS): 2(9) – 3(-2) + 5 18 + 6 + 5 29

Commentary: Remember to square the value of 'a' first before multiplying by

2. Pay attention to the negative signs.

Question 3: Solve the equation: 5m + 8 = 23 Solution: Step 1: Subtract 8 from both sides: 5m + 8 – 8 = 23 – 8 5m = 15 Step 2: Divide both sides by 5: 5m / 5 = 15 / 5 m = 3

Commentary: Always perform the same operation on both sides to maintain the balance of the equation.

Question 4: Expand and simplify the expression: 2(x – 4) + 3(2x + 1)

Solution: Step 1: Distribute the 2 into the first bracket: 2x – 8 Step 2: Distribute the 3 into the second bracket: 6x + 3 Step 3: Combine the two expanded expressions: 2x – 8 + 6x + 3 Step 4: Group like terms: (2x + 6x) + (-8 + 3)* Step 5: Simplify: 8x – 5

Commentary: Carefully track the positive and negative signs during the expansion and simplification process. Independent Practice (Questions Only)

Simplify: 8p – 3q + 5p + q – 7 Evaluate x² + 4xy – y² when x = -1 and y =

2. Solve for k: 3k – 7 = 14 Solve for n: 7n + 5 = 2n - 10 Expand and simplify: 4(y + 3) – (2y – 5)

Expand and simplify: -3(a - 2) + 5(2a + 4)

Solve for x: 5(x - 2) = 3(x + 4) A cellphone costs x rands. A charger costs R50 less than the cellphone. Write an expression for the total cost of the cellphone and the charger, and then simplify the expression.

Solve for y: (y + 3)/2 = 5 Sipho earns m rands per hour.