Introduction to algebra

Letters stand for numbers

In algebra a letter stands for a number. Sometimes it is one particular number we do not know yet; sometimes it can be any number at all. Either way, the letter obeys every rule that numbers obey, which is why algebra works.

Suppose a bag holds $b$ sweets. If you do not know how many there are, you still know that two bags hold $2b$ sweets and that two bags with 3 sweets eaten leave $2b-3$. Whatever $b$ turns out to be, those statements stay true. That is the whole point: one line of algebra says something about every possible value at once.

The words you need:

• A term is a number, a letter, or numbers and letters multiplied together, such as $5$, $x$, $3ab$ or $4x^2$.
• An expression is one or more terms joined by $+$ or $-$, such as $3x+2y-7$. It has no equals sign.
• A formula is a rule connecting two or more quantities, such as $A=lw$ for the area of a rectangle. You put numbers in to get a number out.
• An equation is true only for particular values, such as $3x+2=14$, which is true only when $x=4$.

Writing algebra the standard way

Cambridge expects algebra written in the usual shorthand. These are conventions, so learn them as rules:

• Leave out the multiplication sign: $3\times a$ is written $3a$, and $a\times b$ is written $ab$.
• Put the number first: write $5y$, not $y5$.
• Letters in a product usually go in alphabetical order: $3ab$ rather than $3ba$. Both are correct, but the first is neater.
• Write $1a$ as just $a$.
• Division is written as a fraction: $x÷4$ is $\frac{x}{4}$.
• A power shows repeated multiplication of the letter: $y\times y=y^2$ and $y\times y\times y=y^3$.

Watch the difference between $2y$ and $y^2$. When $y=5$, $2y=10$ but $y^2=25$. In the same way, $3x^2$ means $3\times x^2$: only the $x$ is squared, not the 3. If you want the 3 squared as well, it must be in a bracket: $(3x{)}^2$.

Writing expressions and formulas from words

Turning words into algebra is mostly careful reading. A good habit is to try the sentence with a real number first, then replace the number with the letter.

"Tom has $t$ marbles and Ana has 4 more than Tom." Try it: if Tom had 10, Ana would have $10+4$. So Ana has $t+4$. If Ana had 4 fewer, she would have $t-4$; if she had 4 times as many, $4t$.

Useful phrases:

• "more than", "increased by", "total" mean add;
• "less than", "fewer", "decreased by", "younger" mean subtract, and the order matters: "5 less than $n$" is $n-5$, not $5-n$;
• "times as many", "each", "per" usually mean multiply;
• "shared equally", "per person" often mean divide.

A formula names the quantity it gives, so it starts with a letter and an equals sign. If a plumber charges a call-out fee of $40 plus $25 per hour, the charge $C$ dollars for $h$ hours is

$$C=40+25h$$

Check it with a value you can do in your head: 2 hours costs $40+50=90$ dollars, which matches the words. Always say what your letters stand for, including units, if the question has not already done so.

Letters as generalised numbers

A letter can stand for every number of a certain kind at once. This lets you prove things about all numbers, not just check a few examples.

If $n$ stands for any whole number:

• $2n$ is always even, because it is 2 multiplied by a whole number;
• $2n+1$ is always odd, because it is one more than an even number;
• $n$, $n+1$ and $n+2$ are three consecutive whole numbers;
• $2n$, $2n+2$ and $2n+4$ are three consecutive even numbers, since even numbers go up in steps of 2;
• $10n$ is always a multiple of 10.

Testing $3+4+5=12$ and $10+11+12=33$ suggests that the sum of three consecutive numbers is a multiple of 3. Algebra proves it for all of them: $n+(n+1)+(n+2)=3n+3=3(n+1)$, which is 3 times a whole number. A list of examples, however long, is not a proof; the algebra is.

Substituting into expressions

To substitute means to replace each letter by its value and then work out the result. Two habits prevent almost every error:

1. Put each value in a bracket when you write it in, especially a negative one.
2. Follow the order of operations: brackets, then powers, then multiplication and division, then addition and subtraction.

Take $x=-2$ and the expression $3x^2$. Write $3\times (-2{)}^2$. The power comes first: $(-2{)}^2=4$. Then $3\times 4=12$.

The two classic wrong answers are $36$, from squaring $3\times -2=-6$ (that would be $(3x{)}^2$), and $-12$, from thinking that a negative number squared stays negative. A square of any non-zero number is positive; a cube of a negative number is negative, since $(-2{)}^3=-8$.

Signs need the same care. With $y=-3$, $5-2y=5-2\times (-3)=5-(-6)=11$. Subtracting a negative is adding.

A fraction line acts like a bracket round the top and round the bottom: work each out completely before dividing.

Substituting into formulas

A formula is substituted into in exactly the same way, but the numbers are often decimals and the answer usually has units. On a calculator paper:

• Write the formula, then the formula with the numbers in, before pressing any keys. Most method marks are for that line.
• Use the bracket keys round a whole numerator, a whole denominator, or anything under a square root, or type the fraction with the fraction key.
• Keep every digit on the calculator and round only the final answer, normally to 3 significant figures unless the question says otherwise. Money is given to 2 decimal places.
• Do a rough check. In $V=\frac{1}{3}\pi r^{2}h$ with $r\approx 4$ and $h\approx 10$, $V\approx \frac{1}{3}\times 3\times 16\times 10=160$, so an answer near 175 is believable and one near 1750 is not.

Sometimes the letter you are asked for is not the subject. Substitute the values you are given, then solve the equation that is left. For $v=u+at$ with $v=20$, $u=5$ and $t=6$: $20=5+6a$, so $6a=15$ and $a=2.5$.