Linear Equations

At a glance 11 formulas

1. 1Two kinds, one total$\text{dearer count} = \dfrac{\text{total cost} - n \times \text{cheap price}}{\text{price gap}}$$n = \text{number of items of both kinds}$
2. 2Transfer to equalise$A - x = B + x \Rightarrow A - B = 2x$$A \text{ gives } x \text{ to } B \text{ and they become equal}$
3. 3What is leftIf named parts are fractions of the whole, the rest is 1 minus their sum, and the whole is the rest's count divided by that fraction.
4. 4Fixed plus per unit$\text{bill} = \text{fixed} + \text{rate} \times \text{units}$$\text{rate} = \dfrac{\text{bill}_2 - \text{bill}_1}{\text{units}_2 - \text{units}_1}$
5. 5EliminationScale one equation so an unknown has the same coefficient in both, then subtract. Remove the unknown the question does not ask for.
6. 6Add them allWhen each equation covers a different group of the unknowns, add them all: each unknown appears equally often, so the grand total comes out first.
7. 7Disguised linear$\dfrac{a}{x} + \dfrac{b}{y} = c \;\to\; au + bv = c$$u = \tfrac{1}{x},\ v = \tfrac{1}{y}$
8. 8Products that cancelWhen $r$ rows of $s$ seats become $r + p$ rows of $s - q$ seats, the total stays $rs$: expand $(r + p)(s - q) = rs$, the $rs$ cancels and a linear equation is left.
9. 9Whole-number answersCounts must be whole: find one solution of $ax + by = c$, then step $x$ up by $b$ and $y$ down by $a$ (after dividing out their HCF).
10. 10One solution$\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$$\text{for } a_1x + b_1y = c_1 \text{ and } a_2x + b_2y = c_2$
11. 11None or infinitely many$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}: \text{ none}$$\text{all three equal: infinitely many } (a_ix + b_iy = c_i)$