 maths > algebra

Arithmetic Properties of Equality

what you'll learn...

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=11/2-0.25×2$$2 + 3 = 11 / 2 - 0.25 \times 2$.

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

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

The equation $2+3=11/2-0.25×2$$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=11/2-0.25×2$$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=11/2-0.25×2$$2 + 3 = 11 / 2 - 0.25 \times 2$.

The left hand side $2+3$$2 + 3$ can be modified into $2+3-1+1$$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=11/2-0.25×2$$2 + 3 - 1 + 1 = 11 / 2 - 0.25 \times 2$

Note: the left hand side $2+3$$2 + 3$ cannot be modified into $2+3+2$$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$$2 = 3 - 1$ and $4=2+2$$4 = 2 + 2$.

It is noted that

The equations can be added into a new equation. $2+4=\left(3-1\right)+\left(2+2\right)$$2 + 4 = \left(3 - 1\right) + \left(2 + 2\right)$ .
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. $2×4=\left(3-1\right)×\left(2+2\right)$$2 \times 4 = \left(3 - 1\right) \times \left(2 + 2\right)$
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$$2 = 3 - 1$ and $4=2+2$$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=\left(3-1\right)+\left(2+2\right)$$2 + 4 = \left(3 - 1\right) + \left(2 + 2\right)$

•  Equations can be multiplied or divided (except for expressions evaluating to $0$$0$). eg: $2×4=\left(3-1\right)×\left(2+2\right)$$2 \times 4 = \left(3 - 1\right) \times \left(2 + 2\right)$

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

→   Numerical Arithmetics

→   Arithmetic Operations and Precedence

→   Properties of Comparison

→   Properties of Multiplication

→   Properties of Exponents

→   Algebraic Expressions

→   Algebraic Equations

→   Algebraic Identities

→   Algebraic Inequations