# What are the properties of equalities and what do they do?

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Equalities:

I do not define equality. In a babalance , there is a situation when both sides weigh the same. Please feel the situation of the equality. Both sides you have masses a and b but weigh the same or equal. We say** a=b. **

Now throught the dicussion below, **a=b.**

Let c be any number. Then a**+c**=b**+c.** That is we can add equals to both sides of equals withou affecting equality. Or we can add same to both sides of an equation and the equation remains an equation.

Similarly a**-c**=b-**c**. An equality (or equation) remains equality by subracting the equals from both sides.

An equality can be multiplied by any equal quantity,k on both sides, and the result is an equality. a*k = b*k

An equality can be divided by an equal quantity on both sides, without affecting the equality.But** zero** is an exception.Dividing by zero to establish anythin is not permitted in maths.

a**/k**=b**/k**.

If a = b are equal, then a**^n** = b**^n**. That is the equal powers of equals are equal . This of course, is a consequence of the previous postulates only.

Finally If a=b, then f(a) = f(b), where f is a function of the arguement and f(x) =g(x) , then f(a)=f(a) =g(a)=f(b). Again this is actually consequence the

These properties of equality are useful in solving an equation or while estiblish the proof of an identity.An identy is a relation or equality true for all values of the variable.

Example:

Solve the equation: The sum five times a quantity and six is 16. What is that quantity. Here the unknown quantiy is can be assume x and rest of the thing is setting up an equation: 5x+6=16.

Subtract 6 from both sides.

5x+6-6 =16-6 . Now simplify both sids.

5x=10

Divide both sides by 5 and simplify:

5x/5=10/2

x=2.

Ex 2:

If 5x^3 -40=0.

Add 40 to both sides:

5x^3-40=0+40. Simplify.

5x^3=40. Divide both sides by 5:

5x^3/5=40/5. Simplify.

x^3=8. Take the power (1/3).Or take cube root.

(x^3)^(1/3) = (8)^(1/3)

x=2

Identity: It is form of equality,true for all values of the variable, whereas an equality is true only for particular values.

Ex

x^2-16=(x+4)(x-4) is an identity and is true for any value of x. Where as, x^2-16=0 is an equation holding good for x=+4 or x=-4.

We can use the properties of equality to establish the proof of an idntity.

Equalities:

I do not define equality. In a babalance , there is a situation when both sides weigh the same. Please feel the situation of the equality. Both sides you have masses a and b but weigh the same or equal. We say** a=b. **

Now throught the dicussion below, **a=b.**

Let c be any number. Then a**+c**=b**+c.** That is we can add equals to both sides of equals withou affecting equality. Or we can add same to both sides of an equation and the equation remains an equation.

Similarly a**-c**=b-**c**. An equality (or equation) remains equality by subracting the equals from both sides.

An equality can be multiplied by any equal quantity,k on both sides, and the result is an equality. a*k = b*k

An equality can be divided by an equal quantity on both sides, without affecting the equality.But** zero** is an exception.Dividing by zero to establish anythin is not permitted in maths.

a**/k**=b**/k**.

If a = b are equal, then a**^n** = b**^n**. That is the equal powers of equals are equal . This of course, is a consequence of the previous postulates only.

Finally If a=b, then f(a) = f(b), where f is a function of the arguement and f(x) =g(x) , then f(a)=f(a) =g(a)=f(b). Again this is actually consequence the

These properties of equality are useful in solving an equation or while estiblish the proof of an identity.An identy is a relation or equality true for all values of the variable.

Example:

Solve the equation: The sum five times a quantity and six is 16. What is that quantity. Here the unknown quantiy is can be assume x and rest of the thing is setting up an equation: 5x+6=16.

Subtract 6 from both sides.

5x+6-6 =16-6 . Now simplify both sids.

5x=10

Divide both sides by 5 and simplify:

5x/5=10/2

x=2.

Ex 2:

If 5x^3 -40=0.

Add 40 to both sides:

5x^3-40=0+40. Simplify.

5x^3=40. Divide both sides by 5:

5x^3/5=40/5. Simplify.

x^3=8. Take the power (1/3).Or take cube root.

(x^3)^(1/3) = (8)^(1/3)

x=2

Identity: It is form of equality,true for all values of the variable, whereas an equality is true only for particular values.

Ex

x^2-16=(x+4)(x-4) is an identity and is true for any value of x. Where as, x^2-16=0 is an equation holding good for x=+4 or x=-4.