Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Simplify algebraic expressions by combining like terms.
• Identify coefficients and variables.
• Solve linear equations in one variable.
• Verify solutions by substitution.
• Use the distributive property to simplify and solve.

Lesson summary

This week, we delve into the fascinating world of algebraic expressions and equations. Algebra is a fundamental building block in mathematics, providing the tools to represent unknown quantities and relationships using symbols. Mastering algebra is crucial for success in higher-level mathematics, science, engineering, and even everyday problem-solving. Think about budgeting your pocket money, calculating discounts at the shops, or even figuring out how long it will take to travel a certain distance – algebra can help with all of these!

Lesson notes

2.1 Algebraic Expressions: An algebraic expression is a combination of variables (letters representing unknown values), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division, exponents).

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

Constant: A number that does not change its value (e.g., 5, -3, 0.25).

Coefficient: The number that multiplies a variable in a term (e.g., in the term 3x, the coefficient is 3).

Like Terms: Terms that have the same variable raised to the same power (e.g., 3x and -5x are like terms, but 3x and 3x² are not).

Simplifying Algebraic Expressions: Simplifying an algebraic expression means combining like terms to make the expression shorter and easier to understand. We do this by adding or subtracting the coefficients of like terms.

Example 1: Simplify the expression: 2x + 3y - x + 5y Identify like terms: 2x and -x are like terms; 3y and 5y are like terms.

Combine like terms: (2x - x) + (3y + 5y) = 1x + 8y = x + 8y Example 2: Simplify the expression: 5a - 2b + 3a + b - 4 Identify like terms: 5a and 3a are like terms; -2b and b are like terms; -4 is a constant term.

Combine like terms: (5a + 3a) + (-2b + b) - 4 = 8a - b - 4 2.2 Algebraic Equations: An algebraic equation is a statement that two algebraic expressions are equal. It contains an equals sign (=).

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

Linear Equation: An equation where the highest power of the variable is 1 (e.g., 2x + 3 = 7).

Solving Linear Equations: To solve a linear equation, we need to isolate the variable on one side of the equation. We do this by performing the same operations on both sides of the equation to maintain the equality. The key is to undo the operations that are being performed on the variable.

Example 3: Solve the equation: x + 5 = 12 Subtract 5 from both sides: x + 5 - 5 = 12 - 5 Simplify: x = 7 Example 4: Solve the equation: 3x - 2 = 10 Add 2 to both sides: 3x - 2 + 2 = 10 + 2 Simplify: 3x = 12 Divide both sides by 3: 3x / 3 = 12 / 3 Simplify: x = 4 Example 5: Solve the equation: x/2 + 1 = 4 Subtract 1 from both sides: x/2 + 1 - 1 = 4 - 1 Simplify: x/2 = 3 Multiply both sides by 2: (x/2) 2 = 3 * 2 Simplify: x = 6 2.3 The Distributive Property: The distributive property states that a(b + c) = ab + ac. This means that you can multiply a number by a sum or difference by multiplying the number by each term inside the parentheses.

Example 6: Simplify and solve the equation: 2(x + 3) = 10 Apply the distributive property: 2 x + 2 * 3 = 10 Simplify: 2x + 6 = 10 Subtract 6 from both sides: 2x + 6 - 6 = 10 - 6 Simplify: 2x = 4 Divide both sides by 2: 2x / 2 = 4 / 2 Simplify: x = 2 Example 7: Simplify and solve the equation: -3(y - 2) = 9 Apply the distributive property: -3 y + (-3) * (-2) = 9 Simplify: -3y + 6 = 9 Subtract 6 from both sides: -3y + 6 - 6 = 9 - 6 Simplify: -3y = 3 Divide both sides by -3: -3y / -3 = 3 / -3 Simplify: y = -1 Guided Practice (With Solutions)

Question 1: Simplify the expression: 4p - 2q + 5p + q - 3 Solution: Identify like terms: 4p and 5p are like terms; -2q and q are like terms; -3 is a constant.

Combine like terms: (4p + 5p) + (-2q + q) - 3 = 9p - q - 3 Question 2: Solve the equation: y - 7 = 3 Solution: Add 7 to both sides: y - 7 + 7 = 3 + 7 Simplify: y = 10 Question 3: Solve the equation: 2z + 5 = 11 Solution: Subtract 5 from both sides: 2z + 5 - 5 = 11 - 5 Simplify: 2z = 6 Divide both sides by 2: 2z / 2 = 6 / 2 Simplify: z = 3 Question 4: Solve the equation: x/3 - 2 = 1 Solution: Add 2 to both sides: x/3 - 2 + 2 = 1 + 2 Simplify: x/3 = 3 Multiply both sides by 3: (x/3) 3 = 3 * 3 Simplify: x = 9 Question 5: Solve the equation: 4(a - 1) = 8 Solution: Apply the distributive property: 4 a + 4 * (-1) = 8 Simplify: 4a - 4 = 8 Add 4 to both sides: 4a - 4 + 4 = 8 + 4 Simplify: 4a = 12 Divide both sides by 4: 4a / 4 = 12 / 4 Simplify: a = 3 Independent Practice (Questions Only)

Simplify: 6x + 2y - 3x + 4y - 1 Simplify: -2a + 5b - a - 3b + 7 Solve: m + 9 = 15 Solve: 5n - 3 = 12 Solve: p/4 + 2 = 6 Solve: 3(q + 2) = 15 Solve: -2(r - 3) = 8 Simplify: 7c - 4d + 2c + d - 6 Solve: x/5 - 1 = 3 Solve: 2(y - 4) = -2