Substituting and expanding

Algebra is arithmetic with some numbers written as letters. Every rule you know for numbers still holds. This lesson covers the three things you do with expressions all the time: substituting, collecting like terms and expanding brackets.

Substituting: put the numbers in

To find the value of an expression, replace each letter with its number, in brackets, then work it out in the usual order: brackets, powers, multiply and divide, then add and subtract.

For $a = 4$ and $b = -6$:

Given a ratio instead of the numbers

Sometimes you’re told only $m : n = 3 : 2$ (or $\frac mn = \frac32$), and asked for the value of an expression. You don’t know $m$ and $n$, and you don’t need to. Write both using one new letter: $m = 3k$ and $n = 2k$. When every term has the same power of the letters, $k$ cancels:

• put in $m = 3k$ and $n = 2k$: ${\frac{m^2 + n^2}{m^2 - n^2} = \frac{9k^2 + 4k^2}{9k^2 - 4k^2}}$;
• collect: ${\frac{13k^2}{5k^2}}$;
• $k^2$ cancels: ${\frac{13}{5}}$.

If two ratios share a letter, such as $\frac pq$ and $\frac qr$, write all three letters in terms of the shared one.

Like terms

Like terms have exactly the same letters, to the same powers.

• $3x$ and $-5x$ are like terms.
• $3x$ and $3x^2$ are not.
• $2ab$ and $5ba$ are, because the order of multiplying doesn’t matter.

You can add or take away like terms, and only like terms:

Writing an expression from words

Word problems often ask you to write the expression yourself. Name each quantity with its letter, then turn the sentence into symbols one piece at a time. Say a worker is paid $r$ naira an hour for 40 hours, then $1\frac12$ times that rate for $h$ more hours. The worker earns

Test your expression with small numbers you can check in your head: with $r = 2$ and $h = 2$, that’s $80 + 6 = 86$ naira.

Expanding brackets

A number or letter in front of a bracket multiplies everything inside it:

• $3(2x - 5) = 6x - 15$;
• $-2(x - 4) = -2x + 8$, because a negative times a negative is positive.

Two brackets multiplied together need every term in the first to multiply every term in the second. The grid keeps track of all four products:

Function notation

A function is a rule, written like $f(x) = x^2 + 1$. The letter in the bracket is the input. $f(3)$ means: put 3 in place of $x$ everywhere in the rule. The input can be an expression too. $f(x - 2)$ means: put $(x - 2)$, in brackets, in place of every $x$.

Sometimes the question runs the other way. Say $f(x + 3) = x^2 - 4x$ and you want $f(5)$. Choose $x$ so that the bracket is 5:

• $x + 3 = 5$ gives $x = 2$;
• so $f(5) = 2^2 - 4(2) = -4$.

The inverse function $f^{-1}$ undoes $f$. Do the opposite steps in the reverse order. Or write $y = f(x)$ and make $x$ the subject. For $f(x) = 2x - 5$:

• $y = 2x - 5$ gives $x = \frac{y + 5}{2}$;
• so $f^{-1}(x) = \frac{x + 5}{2}$.

A past question, step by step

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