From Zero to Engineer

Algebra: the laws of algebra, like terms and removing brackets

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}

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.

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Arithmetic - review exercises III

Algebra: review exercises and the laws of exponents