Equal expressions – identities

Algebra, grade 7 Lesson 12 of 22Learning

Equal expressions – identities

Some expressions look different but always give the same value. For example $\displaystyle x+x+x$ and $\displaystyle 3x$ : whatever number we substitute for $\displaystyle x$ , both give the same result.

A table of values

To compare expressions, substitute a few numbers in them and write the values in a table:

$\displaystyle x$	$\displaystyle -2$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 5$
$\displaystyle x+x+x$	$\displaystyle -6$	$\displaystyle 0$	$\displaystyle 3$	$\displaystyle 15$
$\displaystyle 3x$	$\displaystyle -6$	$\displaystyle 0$	$\displaystyle 3$	$\displaystyle 15$
$\displaystyle x+3$	$\displaystyle 1$	$\displaystyle 3$	$\displaystyle 4$	$\displaystyle 8$

For every substitution, $\displaystyle x+x+x$ and $\displaystyle 3x$ are equal. $\displaystyle x+3$ differs from them at the very first one.

An identity

An identity is an equality that holds for every substitution.

Two expressions that are equal for every substitution are called equivalent expressions, and the equality between them is called an identity: $\displaystyle {x+x+x}={3x}$ is an identity. More identities: $\displaystyle {a+b}={b+a}$ , $\displaystyle {2\cdot a\cdot3}={6a}$ , $\displaystyle {x\cdot x}={x^2}$ . Identities are proved with the laws of arithmetic.

A counterexample

A few substitutions where the values agree are still no proof – there may be another one where they differ.

To show that two expressions are not equivalent, one substitution where their values differ is enough. Such a substitution is called a counterexample. For example $\displaystyle 2x$ and $\displaystyle x^2$ : for $\displaystyle {x}={2}$ both are 4, but for $\displaystyle {x}={3}$ , $\displaystyle 2x$ is 6 and $\displaystyle x^2$ is 9. So $\displaystyle {2x}={x^2}$ is not an identity.

Exercise 1

Make a table of values for the expressions $\displaystyle 2(x+1)$ and $\displaystyle 2x+2$ , with $\displaystyle {x}={-3}$ , $\displaystyle {x}={0}$ , $\displaystyle {x}={2}$ and $\displaystyle {x}={10}$ .
Do the expressions seem equivalent?

Exercise 2

For each pair of expressions, find a counterexample that shows they are not equivalent: a. $\displaystyle x+5$ and $\displaystyle 5x$ ; b. $\displaystyle x\cdot x\cdot x$ and $\displaystyle 3x$ ; c. $\displaystyle 10-x$ and $\displaystyle x-10$ .

Exercise 3

Which of the equalities are identities?
For each one that is not, give a counterexample.
a. $\displaystyle {a\cdot b}={b\cdot a}$ ; b. $\displaystyle {a-b}={b-a}$ ; c. $\displaystyle {x+x}={x^2}$ ; d. $\displaystyle {4\cdot x\cdot2}={8x}$ ; e. $\displaystyle {y:1}={y}$ .

Exercise 4

Write a shorter equivalent expression.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: algebra, grade 7

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Adding and subtracting signed numbers

The sum of two opposite numbers
$\displaystyle (-a)+a$ $\displaystyle =$ $\displaystyle 0$
Subtracting a number is adding its opposite
$\displaystyle a-b$ $\displaystyle =$ $\displaystyle a+(-b)$
Subtracting a negative number
$\displaystyle a-(-b)$ $\displaystyle =$ $\displaystyle a+b$

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

A power – $\displaystyle a$ the base, $\displaystyle n$ the exponent: how many times $\displaystyle a$ is multiplied
$\displaystyle a^{n}$ $\displaystyle =$ $\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}$
A number to the power 1 is the number itself
$\displaystyle a^{1}$ $\displaystyle =$ $\displaystyle a$
A negative base with an even exponent – the power is positive
$\displaystyle (-a)^{2}$ $\displaystyle =$ $\displaystyle a^{2}$
The square root – the positive number (or 0) whose square is the number under the root
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The laws of arithmetic and like terms

The commutative law of addition
$\displaystyle a+b$ $\displaystyle =$ $\displaystyle b+a$
The commutative law of multiplication
$\displaystyle a\cdot b$ $\displaystyle =$ $\displaystyle b\cdot a$
The associative law of addition
$\displaystyle (a+b)+c$ $\displaystyle =$ $\displaystyle a+(b+c)$
The associative law of multiplication
$\displaystyle (a\cdot b)\cdot c$ $\displaystyle =$ $\displaystyle a\cdot(b\cdot c)$
A number times itself – squared
$\displaystyle x\cdot x$ $\displaystyle =$ $\displaystyle x^{2}$
Collecting like terms – add the coefficients; the letters stay
$\displaystyle ax+bx$ $\displaystyle =$ $\displaystyle (a+b)x$

The distributive law

The distributive law
$\displaystyle a(b+c)$ $\displaystyle =$ $\displaystyle ab+ac$
The distributive law over a difference
$\displaystyle a(b-c)$ $\displaystyle =$ $\displaystyle ab-ac$
A minus before brackets – every term inside changes its sign
$\displaystyle -(a-b)$ $\displaystyle =$ $\displaystyle -a+b$

Distance, speed and time

$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$