In A Math, simultaneous equations are usually solved by substitution: make $x$ or $y$ the subject of the simpler equation, then sub it into the other one.

(E Math's elimination method rarely works here, because A Math pairs usually have a square or an $xy$ term that adding or subtracting cannot cancel.)

Reflex: Make $x$ or $y$ the subject of the simpler equation, then substitute.

Each solution $(x,y)$ is where graphs meet. Two lines meet once; a line and a curve (or two curves) can meet twice — that's why a quadratic pops up with two answers.

Reflex: Two answers is not a mistake — a line can cut a curve twice.

Make x or y the subject of the line, sub into the curve to get a quadratic. Move all terms to one side, factorize, solve for x.

Reflex: Two x's mean two y's — pair each x with its y before you're done.

If an equation has fractions in it, don't substitute straight away — the algebra turns messy fast.

First multiply every term by the denominators so the fractions cancel:

$$\frac{2}{x} + \frac{3}{y} = 11 \ \xrightarrow{\ \times \ xy\ } \ 2y + 3x = 11xy$$

Here every term was multiplied by $xy$: the $x$ under the 2 cancels, the $y$ under the 3 cancels, and the equation is now fraction-free. Then substitute as usual.

Reflex: See fractions? Multiply them away first, then substitute.

This one is for application questions — word problems that give you a story, not equations. Forming the equations is your job.

Start by naming the two unknowns (for example, let the mass of an apple be $x$ g and the mass of an orange be $y$ g). Then turn each clue in the story into one equation — two unknowns need two clues. Solve the pair, then check both answers against the story and reject anything impossible, like a negative mass.

Reflex: No equations given? Name the two unknowns, turn each clue into an equation, then solve.