Quick questions on Algebraic techniques for HSC Maths Advanced: expanding, factoring, simplifying algebraic fractions and solving linear and simultaneous equations

Factoring reverses expanding, and it is the skill the rest of the course depends on most. There are four methods, and the order matters: always take out the highest common factor first, then pick the method by counting the terms.

An algebraic fraction is just a fraction with pronumerals, and the arithmetic rules carry over unchanged, with factoring doing most of the work.

The governing principle is simple: do the same thing to both sides to keep the equation balanced. You may add or subtract any number from both sides, and multiply or divide both sides by any non-zero number. Work towards isolating the unknown.

A literal equation is one where you rearrange to make a chosen pronumeral the subject, treating the other letters as constants. The same balancing moves apply. Where the target pronumeral appears in more than one term, gather those terms on one side and factor it out, then divide. For example, to make $x$ the subject of $y = \dfrac{x + 1}{x + 2}$: multiply out to $xy + 2y = x + 1$, gather $xy - x = 1 - 2y$, factor $x(y - 1) = 1 - 2y$, and divide to get $x = \dfrac{1 - 2y}{y - 1}$.

Two linear equations in two unknowns usually have one solution: the single pair $(x, y)$ that satisfies both. There are two algebraic methods, and you choose whichever is tidier.

The cleanest way to see what "solving simultaneously" means is to graph both lines and watch them meet. Below, the pair $3x + 2y = 12$ and $5x - 2y = 4$ is built up one line at a time; the meeting point is the algebraic solution $x = 2$, $y = 3$. (This is the same worked example solved by elimination below.)

A sign slip during elimination is easy to make and easy to catch: substitute your $(x, y)$ back into both originals. Markers often award a mark for a correct check, and it costs seconds.