overview

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 equalities in algebra.*

to equate means to declare as equal

An equation is a "statement of two quantities being equal".

Consider the equation $2+3=11/2-0.25\times 2$.

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

The right hand side of the equation is $11/2-0.25\times 2$.

The equation $2+3=11/2-0.25\times 2$ states that the left hand side equals the right hand side.

changing an equation

Consider the equation $2+3=11/2-0.25\times 2$.

The left hand side can be modified without affecting the statement of equality. The left hand side is a numerical expression which can be modified as per PEMA Precedence / CADI Laws and Properties of Arithmetics.

The same for right hand side of the equation too.

Considering the equation $2+3=11/2-0.25\times 2$.

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 $2+3-1+1=11/2-0.25\times 2$

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.

two equations

Consider the two equations $2=3-1$ and $4=2+2$.

It is noted that

The equations can be added into a new equation. $2+4=(3-1)+(2+2)$ .

The statement of left-hand-side being equal to right-hand-side holds in the new equation after the addition.

The equations can be multiplied into a new equation. $2\times 4=(3-1)\times (2+2)$

The statement of left-hand-side being equal to right-hand-side holds in the new equation after the multiplication.

Considering 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: $2\times 4=(3-1)\times (2+2)$

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

summary

** 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. *

Outline

The outline of material to learn "Algebra Foundation" is as follows.

Note: *click here for detailed outline of Foundation of Algebra*

→ __Numerical Arithmetics__

→ __Arithmetic Operations and Precedence__

→ __Properties of Comparison__

→ __Properties of Addition__

→ __Properties of Multiplication__

→ __Properties of Exponents__

→ __Algebraic Expressions__

→ __Algebraic Equations__

→ __Algebraic Identities__

→ __Algebraic Inequations__

→ __Brief about Algebra__