Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Simplify algebraic expressions.
• Solve linear equations and inequalities.
• Factorise quadratic expressions.
• Solve quadratic equations by factorisation.
• Apply algebraic skills to solve word problems.

Lesson summary

This week's revision focuses on key concepts covered in the past nine weeks. This is crucial as a strong foundation in these areas is essential for success in Grade 10 Mathematics and future studies. Specifically, we will be reinforcing skills in algebraic expressions, equations, and inequalities, which are fundamental to problem-solving in mathematics and many real-world scenarios. These concepts are vital for understanding finances (budgeting, investments), calculating construction measurements, and interpreting data in scientific research – all important skills needed in South Africa.

Lesson notes

2.1 Algebraic Expressions An algebraic expression is a combination of variables (e.g., x, y, a), constants (e.g., 2, -5, ½) and mathematical operations (addition, subtraction, multiplication, division, exponentiation).

Simplifying Algebraic Expressions: Expanding Brackets: Use the distributive property: a(b + c) = ab + ac.

Example: 3(x + 2) = 3x + 6 Combining Like Terms: Like terms have the same variable raised to the same power. Combine their coefficients.

Example: 2x + 5x = 7x

Example:** Simplify the expression: 2(x + 3) – (x – 1)

Solution: Expand the brackets: 2x + 6 – x + 1 (Note the distribution of the negative sign!)

Combine like terms: (2x – x) + (6 + 1)

Simplified expression: x + 7 2.2 Equations An equation is a statement that two expressions are equal. Solving an equation means finding the value(s) of the variable(s) that make the equation true.

Solving Linear Equations: Isolate the variable by performing the same operations on both sides of the equation. Remember to maintain the equality.

Example:** Solve for x: 2x + 3 = 7 Solution: Subtract 3 from both sides: 2x + 3 - 3 = 7 - 3 Simplify: 2x = 4 Divide both sides by 2: 2x / 2 = 4 / 2 Solution: x = 2 Solving Linear Equations with Fractions: Find the lowest common denominator (LCD) of all fractions in the equation. Multiply both sides of the equation by the LCD to eliminate the fractions. Solve the resulting equation as usual.

Example: Solve for y: y/2 + 1/3 = 5/6 Solution: The LCD of 2, 3, and 6 is

6. Multiply both sides by 6: 6(y/2 + 1/3) = 6(5/6)

Distribute: 3y + 2 = 5 Subtract 2 from both sides: 3y = 3 Divide by 3: y = 1 2.3 Inequalities An inequality is a statement that compares two expressions using inequality symbols: (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

Solving Linear Inequalities: Solve inequalities in the same way as equations, with one important exception: When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.

Example: Solve for x: -2x + 4 > 10 Solution: Subtract 4 from both sides: -2x > 6 Divide both sides by -2 (and reverse the inequality sign): x 4 Solve for b: 3(b + 2) ≤ 15 Factorise: x² + 8x + 15 Factorise: y² - 25 Solve for x: x² + 6x + 8 = 0 Solve for p: 2p² + 5p + 2 = 0