Algebra Notes

Algebra Overview

• Transposing (Changing the Subject of the Formula)

  • Formula: $V = \frac{1}{3} \pi r^2 h$

  • Solving for $r$ when $V = 44$, $h = 7$, and $\pi = \frac{22}{7}$

Practice Transposing

• Transposing Examples

  • Given formulas to rearrange.

Equations

• Types of Equations to Solve

  • $2(x - 5) = 12$

  • $3x + 7 = -11 + 2x$

  • $-18 - 6k = 6(1 + 3k)$

  • $-1(1 + 7x) - 6(-7 - x) = 36$

Inequalities

• Understanding Inequalities

  • Example: $3p - 4 \leq p + 6$

  • Rearranging yields: $2p \leq 10 \Rightarrow p \leq 5$

Solving Inequalities

• Example Steps:

  • Solve: 5x + 2 > 3x + 14

  • Rearranging: 5x - 3x > 14 - 2

  • Result: 2x > 12 \Rightarrow x > 6

Important Notes on Inequalities

• When multiplying/dividing by a negative number, reverse the inequality sign.

Visual Representation of Inequalities on the Number Line

• Open Circle: Represent values not included (example: x > 2).

• Filled Circle: Represent values included (example: $x \geq 3$).

• Between Two Values: Two circles are used to represent ranges.

Simultaneous Equations Using the Elimination Method

• Example Set:

  • $x + y = 4$

  • $3x + y = 7$

  • Re-arranging gives:

  • Eliminating y.

  • Substitute back for solutions.

More Examples

• Solve forms such as:

  • $2x + 5y = 16$

  • $3x - 2y = 5$

Practice Questions

• Include equations and inequalities to solve:

  • $4x - 8 = -12$

  • Evaluating expressions given values (e.g., $a = -1, b = 3, c = -4$).

Final Notes

• Understanding Symbols:

  • Closed Circle for included values, Open Circle for excluded values on a number line.

• Key Takeaway: Always remember to reverse inequality signs when necessary during solving processes and to practice on various forms and styles for comprehensive understanding.

Practice Problems Given

• Various forms from basic equations to inequalities and their representation.