Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Simplify algebraic expressions by combining like terms.
• Evaluate algebraic expressions by substituting values for variables.
• Solve simple one-step equations using inverse operations.

Lesson summary

This week, we delve deeper into the fascinating world of algebraic expressions and simple equations. Algebra is a fundamental building block in mathematics and is crucial for solving everyday problems. Understanding algebra allows us to represent unknown quantities, make predictions, and solve problems involving relationships between different things. In South Africa, these skills are vital for various fields, from managing personal finances to understanding complex economic models, designing infrastructure, and even in sports analysis.

Lesson notes

2.1 Algebraic Expressions: An algebraic expression is a combination of numbers, variables (letters representing unknown values), and operation symbols (+, -, ×, ÷).

Variable: A symbol (usually a letter like x, y, or a) that represents an unknown number.

Constant: A number that does not change its value.

Example: 5, -3, 1/

2. Coefficient: The numerical factor of a term that contains a variable. In the term 5x, the coefficient is

5. If a term is just 'x' the coefficient is understood to be

1. Like Terms: Terms that have the same variable(s) raised to the same power.

Example: 3x and 5x are like terms. 2x² and -x² are like terms. 3x and 3x² are NOT like terms because the power of x is different. 2xy and 5yx are like terms because they contain the same variables to the same power (order of multiplication doesn't matter).

Example 1: Identifying Parts of an Algebraic Expression Consider the expression: 7x + 3y - 5 + 2x² - y/2 Variables: x, y Constants: -5 Terms: 7x, 3y, -5, 2x², -y/2 Coefficients: 7 (for the term 7x), 3 (for the term 3y), 2 (for the term 2x²), -1/2 (for the term -y/2) 2.2 Simplifying Algebraic Expressions by Combining Like Terms Simplifying an expression means rewriting it in a shorter, easier-to-understand form. We can only combine like terms.

Example 2: Simplifying Algebraic Expressions Simplify: 4a + 7b - 2a + b - 3 Identify like terms: (4a and -2a) are like terms, (7b and b) are like terms, and -3 is a constant term.

Combine like terms: 4a - 2a = 2a 7b + b = 8b (Remember b is the same as 1b)

Write the simplified expression: 2a + 8b - 3 Example 3: More Complex Simplification Simplify: 5x² - 3x + 2 + x² + 4x - 7 Identify like terms: (5x² and x²), (-3x and 4x), (2 and -7)

Combine like terms: 5x² + x² = 6x² -3x + 4x = x 2 - 7 = -5 Write the simplified expression: 6x² + x - 5 2.3 Evaluating Algebraic Expressions Evaluating an expression means finding its numerical value by substituting given values for the variables.

Example 4: Evaluating an Expression Evaluate the expression 3x + 2y - 5 when x = 4 and y = -1 Substitute the given values: 3(4) + 2(-1) - 5 Perform the multiplication: 12 - 2 - 5 Perform the addition and subtraction: 12 - 2 - 5 = 5 Example 5: Evaluating a More Complex Expression Evaluate the expression x² - 4xy + y² when x = 2 and y = -3 Substitute the given values: (2)² - 4(2)(-3) + (-3)² Calculate the exponents: 4 - 4(2)(-3) + 9 Perform the multiplication: 4 - (-24) + 9 Perform the addition and subtraction: 4 + 24 + 9 = 37 2.4 Simple Equations: An equation is a statement that two expressions are equal. It always contains an equal sign (=). Solving an equation means finding the value of the variable that makes the equation true. 2.5 Solving One-Step Equations using Inverse Operations To solve an equation, we need to isolate the variable on one side of the equation. We do this by performing inverse operations. Inverse operations "undo" each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. The key is to do the same operation on both sides of the equation to keep it balanced. Think of it like a scale; if you add weight to one side, you must add the same weight to the other side to keep it level.

Example 6: Solving a One-Step Equation (Addition)

Solve: x + 5 = 12 Identify the operation being done to the variable: 5 is being added to x. Perform the inverse operation on both sides: Subtract 5 from both sides: x + 5 - 5 = 12 - 5 Simplify: x = 7 Example 7: Solving a One-Step Equation (Subtraction)

Solve: y - 3 = 8 Identify the operation being done to the variable: 3 is being subtracted from y. Perform the inverse operation on both sides: Add 3 to both sides: y - 3 + 3 = 8 + 3 Simplify: y = 11 Example 8: Solving a One-Step Equation (Multiplication)

Solve: 4z = 20 Identify the operation being done to the variable: z is being multiplied by

4. Perform the inverse operation on both sides: Divide both sides by 4: 4z / 4 = 20 / 4 Simplify: z = 5 Example 9: Solving a One-Step Equation (Division)

Solve: a / 2 = 6 Identify the operation being done to the variable: a is being divided by

2. Perform the inverse operation on both sides: Multiply both sides by 2: (a / 2) 2 = 6 2 Simplify: a = 12 2.6 Solving Two-Step Equations using Inverse Operations Two-step equations require two inverse operations to isolate the variable. Remember to perform operations in reverse order of operations (PEMDAS/BODMAS).

Therefore, we typically undo addition/subtraction before multiplication/division.