Algebra essentials

12a³

HintMultiply the plain numbers together, then count how many of the letter are being multiplied in all.

The whyThe numbers give 4 × 3 = 12, and a × a² is a × a × a, which is a³, so the answer is 12a³. Multiplying powers of the same letter adds the indices, because you are simply counting how many copies sit in the product.

6

HintCall the starting number n, write both descriptions of what comes out, and set them equal.

The whyThe output is 2n + 6 and also 3n, so 2n + 6 = 3n and n = 6. Checking: doubling 6 and adding 6 gives 18, which is three times 6. Turning a machine into an equation is how function machines lead into algebra.

17

HintSwap the letter for its number, then work out the multiplication before the addition.

The whySubstituting turns the expression into 3 × 5 + 2. The order of operations still applies once the letter has gone, so the multiplication happens first.

8a

HintBoth parts are counting the same letter, so add up how many there are.

The whyFive lots of a plus three lots of a makes eight lots of a. Only terms with an identical letter part can be gathered together like this.

2(n + 5), which expands to 2n + 10

HintPut the increase inside brackets first, so the doubling acts on the whole of it.

The whyThe brackets record the order of events: add 5, then double everything that resulted. Without them, 2n + 5 would double only the original number. Expanding the brackets gives 2n + 10 — the same expression written another way, so that form is right too.

An equation

HintIt has a balance point in the middle, and your job is to find the value that keeps both sides level.

The whyAn expression such as 3x + 2 has no equals sign, so it can only be simplified or evaluated. Once an equals sign joins two expressions, there is something to solve.

3x + 12

HintThe number outside must reach every term inside — nothing may be left behind.

The whyMultiply the 3 by each term in turn: 3 × x and 3 × 4. This is the distributive law, the same rule that makes 3 × 24 easier as 3 × 20 + 3 × 4.

2x + 4y

HintSort the terms into piles by their letter, then count each pile separately.

The whyThe x terms give 4x − 2x and the y terms give 3y + y, where a lone y counts as 1y. Unlike terms stay apart because they are counting different things.

3x = 15, so x = 5

HintWhatever you do to one side, do to the other — the balance must stay level at every step.

The whyThe two inverse steps come in reverse order: the + 4 was applied last, so it is undone first. Substituting the answer back into 3x + 4 = 19 is the quickest way to check it.

3n + 2

HintThe gap between terms tells you what multiplies the position number; then adjust to land on the opening term.

The whyThe common difference is 3, so the rule begins 3n. Since 3 × 1 = 3 but the sequence starts at 5, a further 2 is added to every term.

y-intercept

HintIt is the height at which the line crosses the vertical axis.

The whyIn y = mx + c the m is the gradient, or steepness, and the c is where the line cuts the vertical axis. Those two numbers pin down a whole straight line.

The solution to both equations

HintThe crossing point belongs to each line at once, so its coordinates fit each rule.

The whyEvery point on a line makes its own rule true, so the meeting point is the one pair of x and y values that works for the pair of lines together. That is what solving simultaneously means.