Algebra essentials

12a³

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

为什么The 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

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

为什么The 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

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

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

8a

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

为什么Five 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

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

为什么The 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

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

为什么An 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

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

为什么Multiply 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

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

为什么The 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

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

为什么The 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

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

为什么The 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

提示It is the height at which the line crosses the vertical axis.

为什么In 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

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

为什么Every 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.