These properties will be necessary to solving equations. They should be committed to memory.
Properties of Equality
Properties of Equality
The first four properties of equality--those that deal with operations--allow us to add, subtract,
multiply and divide variables. They also formally express the idea that when we perform the same
operations on both sides of an equation, the two sides are still equivalent. In other words, that we
can perform the same operation on both sides without changing the values of the variables.
The next four properties formally express the idea that the quantities on each side of the "=" sign
mean the same thing as each other, and that is does not matter which quantity is on which side of
the "=" sign.
The final property--the Zero Product Property--is the inverse of the Multiplication Property of Zero
learned in Pre-Algebra. The Multiplication Property of Zero states that the product of several
numbers, one which is zero, is zero. The Zero Product Property states that if the product of several
numbers is zero, then at least one of them must be zero--we cannot multiply non-zero numbers and
get zero.
Properties of Operations and Identities
Properties of Operations and Identities
The properties of operations and identities formally express the properties learned in the Operations
section of the Pre-Algebra SparkNote, with two additions--the Quotient Property and the
Multiplicative Inverse Property. The Quotient Property stems from the fact that we multiply two
fractions by multiplying their numerators and then multiplying their denominators. When we multiply
a by
, the numerator is a×1 = a and the denominator is 1×b = b ; thus, the result is
. The
Quotient Property shows that multiplying by a fraction whose numerator is 1 is the same as dividing
by the denominator of that fraction, and that dividing by a number is the same as putting the divisor
in the denominator of a fraction whose numerator is the given number.
We can see how the Multiplicative Inverse Property works using the Quotient Property. We want to
find a number that yields 1 when multiplied by a , or that yields
Thus, we have a×
=
, where
when multiplied by a , since
represents the fraction we are trying to find. Using the Quotient
Property, we can replace b with a to yield the desired equation -- thus,
a in the denominator:
=
=1.
. The number
has 1 in the numerator and
is called the reciprocal or multiplicative inverse of a . If
a is a fraction, then we can find the reciprocal of a by switching the numerator and the denominator.
Example: Name the property or operation used in each step when solving 206 = 11x + 96 :
1.
2.
3.
4.
206 = 11x + 96Given
206 - 96 = 11x + 96 - 96Subtraction Property of Equality
110 = 11x + 0Subtraction
110 = 11xAdditive Identity Property
5.
=
Division Property of Equality
6. 10 = 1xDivision
7. 10 = xMultiplicative Identity Property
8. x = 10Symmetric Property