What is an equation? . . . . .What is NOT an equation?

Before starting, a definition: Any combination of numbers and letters and arithmetical operations (including   =   <   >   <=   >=) with more than three symbols is an algebra “thing”.  So, in passing, observe that a “number sentence” is an algebra “thing”.

Equations are neither true nor false: Some examples –

x + 10 = 45
x + 2y = 8
x^2 + y^2 = 4
y = x^2 + 5x + 7
x^2 + 5x + 7 = 0
x = 35
ax + by + c = 0

In each case the equation specifies the value or values of the letter quantities

x + 10 = 45
The value of x is such that if I add 10 to it I get 45

x + 2y = 8
The values of x and y are such that twice the y value added to the x value gives me 8

x^2 + y^2 = 4
The values of x and y are such that the square of the x value added to the square of the y value is equal to 4. This and the one above specify pairs of values.

y = x^2 + 5x + 7      You do these two
x^2 + 5x + 7 = 0

x = 35
The value of x is specified to be 35
and lastly,   ax + by + c = 0
————————————-
Then we have identities, sometimes called equivalence statements.
These are ALWAYS true.
Examples:

3 + 2 = 5
4 = 1 + 3
8 = 11 – 3
(x + 1)^2 = x^2 + 2x + 1
2x(4 + 7) = 2×4 + 2×7 (where x is multiplied by)
6/8 = 3/4
————————————-
Later on, in algebra, we get definitions:

f(x) = 3x + 2
This means “The rule for the function whose name is f and whose input is x is multiplythevalueofhteinputxbythreeandaddtwotoit”
or “The value of the output of the function f for input x is the value of 3x + 2”.

y = f(x)
This means “The value of the output of the function f for input x is to be given the name y”.
These are NOT equations and they are NOT identities.
————————————-
The whole current mess arises from the use of the equals sign for “gives” or “makes”, or “we get”, as in “3 + 5 makes 8”, or  “If we multiply 4 by 6 we get 24”, and we write

3 + 5 = 8, and 4 x 6 = 24
3 + 5 is 8 and 4 x 6 is 24 would be better.

The newfangled term “number sentence” appears to have been invented in order to avoid dealing with the correct mathematical jargon, but I see it as making things EVEN worse,

 

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