Equations

The ancient art of balancing

• The word "equation" comes from "equal" — both sides must be the same, like a balance scale.
• Babylonians solved linear equations 一次方程 4000 years ago using clay tablets. Today we use the same principle: do the same to both sides.

Linear equations

• Solve by doing the same operation to both sides until the unknown is alone.
• $3x + 7 = 22 \Rightarrow 3x = 15 \Rightarrow x = 5$.

Solve by doing the same to both sides

Simultaneous equations 联立方程

• Two equations, two unknowns. Eliminate 消元 one variable by adding or subtracting.
• $2x + y = 7$ and $3x - y = 8$: adding gives $5x = 15$, so $x = 3$, then $y = 1$.

Quadratic 二次 equations (Extended)

• A quadratic $ax^2 + bx + c = 0$ can have two, one, or zero real solutions.
• By factorising: $x^2 + 5x + 6 = 0 \Rightarrow (x+2)(x+3) = 0 \Rightarrow x = -2$ or $x = -3$.
• By the formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (given in the exam).

The roots of a quadratic are where the parabola crosses the $x$-axis — the solutions of $y = 0$.

Rearranging formulae

• Make a letter the subject 主项 by reversing the operations applied to it.
• Make $r$ the subject of $V = \dfrac{4}{3}\pi r^3$: $\;r^3 = \dfrac{3V}{4\pi}$, so $r = \sqrt[3]{\dfrac{3V}{4\pi}}$.

You've got it