Algebra essentials

3 multiplied by x

HintThe number sits right next to the letter with no sign between them — what operation is hiding there?

The whyWriting 3x instead of 3 × x keeps the multiplication sign from being confused with the letter x. The number in front is called the coefficient.

13

HintFollow the two steps in the order the machine sets them out.

The whyMultiplying first gives 12, and adding 1 finishes the job. Running the steps the other way round would give 16, so the order a machine works in matters.

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)

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.

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.

x = 5

HintUndo the operations in reverse order — deal with what was added before what multiplies.

The whySubtracting 4 from both sides gives 3x = 15, and dividing both sides by 3 finishes it. Whatever you do to one side must be done to the other to keep the balance.

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.