Letters in place of numbers, the names of the operations, the laws of commutativity, associativity and distributivity with counterexamples, and the basic manipulations: combining like terms, factoring out the common factor and removing brackets.

Algebra begins where we write a letter instead of a particular number. The rules stay the same as in arithmetic - what changes is that they hold for every number at once. This lesson sets out the three laws everything else rests on: commutativity, associativity and distributivity. Just as important is what they do not say: subtraction and division are neither commutative nor associative, and division is distributive from one side only.

Algebra

a + a + a + a = 4a \quad (4 \cdot a)

3a - a = 2a

8a : a = 8

a \cdot a \cdot a \cdot a \cdot a = a^{5}

a + b

the sum of a and b

a - b

the difference between a and b

ab

the product of a and b

\displaystyle a : b, \quad a/b, \quad \frac{a}{b}

the quotient of a and b, where b ≠ 0

The laws of algebra

1) Commutativity

x + y = y + x

xy = yx

Addition and multiplication are commutative operations.

x - y \neq y - x

x : y \neq y : x

\displaystyle \frac{x}{y} \neq \frac{y}{x}

Subtraction and division are not commutative operations.

2) Associativity

x + (y + z) = (x + y) + z = x + y + z

x(yz) = (xy)z = xyz

Addition and multiplication are associative operations.

x - (y - z) \neq (x - y) - z

x : (y : z) \neq (x : y) : z

Subtraction and division are not associative operations.

3) Distributivity

x(y + z) = xy + xz

(x + y)z = xz + yz

x(y - z) = xy - xz

(x - y)z = xz - yz

Multiplication is distributive over addition and subtraction, both from the left and from the right.

(x + y) : z = x : z + y : z

x : (y + z) \neq x : y + x : z

\displaystyle \frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

\displaystyle \frac{x}{y + z} \neq \frac{x}{y} + \frac{x}{z}

Division is distributive over addition and subtraction, but only from the right.

Combining like terms

4x + 3y - 2z + 5y - 3x + 4z = x + 8y + 2z

4uv - 7uz - 6wz + 2uv + 3wz = 6uv - 7uz - 3wz

Factoring out the common factor

9st - 3sv - 6sw = 3s(3t - v - 2w)

Removing brackets

\begin{aligned}\frac{y + x}{8x} - \frac{y - x}{4x} &= \frac{y}{8x} + \frac{x}{8x} - \frac{y}{4x} + \frac{x}{4x} \\&= \frac{y}{8x} + \frac{1}{8} - \frac{y}{4x} + \frac{1}{4} = \frac{3}{8} - \frac{y}{8x}\end{aligned}

\displaystyle \frac{1}{4} = \frac{2}{8}

\displaystyle \frac{1}{8} + \frac{2}{8} = \frac{3}{8}

\displaystyle \frac{y}{4x} = \frac{2y}{8x}

\displaystyle \frac{y}{8x} - \frac{2y}{8x} = -\frac{y}{8x}

\begin{aligned}4[2x + 3[5 - 2(x - y)]] &= 4[2x + 3[5 - 2x + 2y]] \\&= 4[2x + 15 - 6x + 6y] \\&= 8x + 60 - 24x + 24y \\&= -16x + 24y + 60\end{aligned}

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Frequently asked questions

What does it mean for an operation to be commutative?

That the order of the numbers does not change the result: x + y = y + x and xy = yx. Subtraction is not commutative - for 3 and 5 you get −2 one way and 2 the other.

Why is division distributive only from the right?

Because dividing a sum by a number can be split term by term: (x + y) : z = x : z + y : z. The other way round fails: x : (y + z) is not the same as x : y + x : z - for 3, 5 and 7 you get 0.25 instead of about 1.03.

What are like terms?

Terms with the same set of letters, for example 4uv and 2uv. Only those may be added or subtracted - you add the coefficients and the letters stay unchanged.

How do you remove brackets nested inside one another?

From the inside out. In 4[2x + 3[5 − 2(x − y)]] you first multiply the innermost bracket by −2, then by 3, and finally by 4. The result is −16x + 24y + 60.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/from-zero-to-engineer/algebra-laws-of-algebra. On the lesson page you will also find games and practice exercises - free and without an account.

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Arithmetic - review exercises III

Algebra: review exercises and the laws of exponents