__maths__>__Foundation of Algebra with Numerical Arithmetics__>__Numerical expressions, equations, identities, and in-equations for Algebra__### Arithmetic Properties of Equality

In this lesson properties of equations are explained.

• equations are statement of equality between two expressions

• the statement of equality remains unchanged when the expressions are modified per PEMA / CADI

• the statement of equality remains unchanged for arithmetics between two equations

It is very important to go through this once to understand in-equalities in algebra.

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What is an equation?

- statement of two quantities being equal
- statement of two quantities being equal
- an imaginary line between north and south pole

The answer is "statement of two quantities being equal".

Consider the equation `2+3 = 11//2 - 0.25xx2`.

The left hand side of the equation is `2+3`.

The right hand side of the equation is `11//2 - 0.25xx2`.

The equation `2+3 = 11//2 - 0.25xx2` states that the left hand side equals the right hand side.

Consider the equation `2+3 = 11//2 - 0.25xx2`.

Can the left hand side be modified without affecting the statement of equality?

- Yes, the left hand side is a numerical expression which can be modified as per PEMA Precedence / CADI Laws and Properties of Arithmetics
- Yes, the left hand side is a numerical expression which can be modified as per PEMA Precedence / CADI Laws and Properties of Arithmetics
- No, the left hand side is not a numerical expression.

The answer is "Yes, the left hand side is an numerical expression which can be modified".

Consider the equation `2+3 = 11//2 - 0.25xx2`.

The left hand side `2+3` can be modified into `2+3-1+1` as per the additive identity and inverse properties. The value of the expression is not changed, so the equation holds true for `2+3-1+1 = 11//2 - 0.25xx2`

Note: the left hand side `2+3` cannot be modified into `2+3+2`, as this changes the value of the expression. That is, the equation will not be true with this modified expression.

Consider the two equations `2=3-1` and `4=2+2`. Which of the following is true?

- The equations can be added into a new equation. `2+4 = (3-1) + (2+2)`
- The equations can be multiplied into a new equation. `2xx4 = (3-1) xx (2+2)`
- both the above
- both the above

The answer is "both the above"

Consider two equations `2=3-1` and `4=2+2`. Multiple equations can be used to arrive at equations derived from them as per the following.

• Equations can be added or subtracted. eg: `2+4 = (3-1) + (2+2)`

• Equations can be multiplied or divided (except for expressions evaluating to `0`). eg: `2xx4 = (3-1) xx (2+2)`

• Equations can be taken exponent of (when expressions evaluate to integers). `2^4 = (3-1)^(2+2)`

** Numerical Equations** :

An equation is a statement of equality between left hand side and right hand side.

Equation consists of expressions on the LHS and RHS. These expressions may be modified as per PEMA precedence / CADI Laws and Properties of Arithmetics.

Multiple equations can be used to arrive at equations derived from them as per the following.

• Equations can be added or subtracted

• Equations can be multiplied or divided (except for expressions evaluating to `0`)

• Equations can be taken exponent of (when expressions evaluate to integers) *Note 1: For expressions evaluating to fractions or decimals, the exponent should be worked out with details. This will be explained in higher classes. Note 2 : The inverses of exponent, that is root and logarithm are explained in higher classes. *

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