Algebra foundations

Pronumerals and conventions

A pronumeral is a letter standing for a number. Algebra hides some symbols: $3 \times a$ is written $3a$, $a \div b$ is written $\frac{a}{b}$, and $a \times a$ is $a^2$. In the term $7a^2b$, the coefficient is 7. An expression is a collection of terms joined by $+$ and $-$ signs — it has no equals sign.

Writing expressions

Exams describe a situation in words and ask for the algebra. Choose a distinct pronumeral for each unknown, then translate phrase by phrase: if Bella collects $b$ dollars, “three times as much” is $3b$ and “thirty dollars less than twice as much” is $2b - 30$. A total is a sum: $b + 3b + (2b - 30) = 6b - 30$. Define your pronumerals in writing before using them — it is often a mark on its own.

Collecting like terms

Like terms have exactly the same pronumerals raised to the same powers; the order of the letters does not matter, so $x^2y$ and $yx^2$ are like terms, but $x^2y$ and $xy$ are not. Add or subtract the coefficients only:

$$2x + 3y + 7x - 12y = 9x - 9y$$

Each sign belongs to the term after it — move terms around with their signs attached.

Multiplying and dividing terms

To multiply, handle numbers first and then each pronumeral: $5ab \times 7a = 35a^2b$. Write divisions as fractions and cancel common factors from top and bottom:

$$\frac{12ab \times 5a}{4a} = 15ab \qquad 5ab \div 25ab^2 = \frac{1}{5b}$$

Unlike collecting, any terms can be multiplied or divided — like or not.

Substitution

Substituting means replacing pronumerals with given values, then following the order of operations. Put negative values in brackets before doing anything else. With $a = -5$ and $b = 2$:

$$3a^2 - 5b = 3 \times (-5)^2 - 5 \times 2 = 75 - 10 = 65$$