are algebraic equations the same no matter what the arrangement is?

[mathstriker]

Take a look at the following equation:

6-7e+9h-2h+5e

Now, this is the simplified version of it:

6-2e+7h

and this is the same version but with another arrangement:

-2e+7h+6

I wanna ask, are they all the same equations? When simplifying an equation, does the order you put the expressions after simplifying matter?

Thank you so much.

 

[Deleted member 4993]

Take a look at the following equation:

6-7e+9h-2h+5e

Now, this is the simplified version of it:

6-2e+7h

and this is the same version but with another arrangement:

-2e+7h+6

I wanna ask, are they all the same equations? When simplifying an equation, does the order you put the expressions after simplifying matter?

Thank you so much.

First thing about mathematical "jargon". Those are NOT equations - equations must contain "=" sign.

Yes - those are all SAME expressions.

Many times we write "one way or the other" to make the "next" step of intended operation easy to follow.

Sometimes those are written in particular order as "personal preference".

 

[Steven G]

What you listed are NOT equations. Equations have equal signs. What you posted are expressions.

Variables are numbers which just are not known at the moment.

If something is true for numbers then it is true for variables.
If something is true for variables then it is true for numbers.
If something is NOT true for numbers then it is NOT true for variables.
If something is NOT true for variables then it is NOT true for numbers.

I hope this answers your question

 

[mathstriker]

First thing about mathematical "jargon". Those are NOT equations - equations must contain "=" sign.

Yes - those are all SAME expressions.

Many times we write "one way or the other" to make the "next" step of intended operation easy to follow.

Sometimes those are written in particular order as "personal preference".

The issue is, I don't how and what order should I follow when simplifying expression like this: 6-7e+9h-2h+5e? I mean how should I write the simplified version? With which expression should I begin?

Thanks.

 

[Deleted member 4993]

The issue is, I don't how and what order should I follow when simplifying expression like this: 6-7e+9h-2h+5e? I mean how should I write the simplified version? With which expression should I begin?

Thanks.

In general:

It does not matter - in what order those elements of expressions are shown.

If you need to evaluate the expression, at given values of e and h, you will need to follow rule of PEMDAS (or BODMAS)

 

[HallsofIvy]

The issue is, I don't how and what order should I follow when simplifying expression like this: 6-7e+9h-2h+5e? I mean how should I write the simplified version? With which expression should I begin?

Thanks.

To write the "simplified version" write it as simply as possible! Do you see that "-7e+ 5e" is the same as "(-7+ 5)e" which is the same as "-2e"? Do you see that "-2e" is simpler than "-7e+ 5e"? Similarly, do you agree that "7h" is equal to "9h- 2h" and is similar? So 6- 7e+ 9h- 2h+ 5e represents the same number as 6- 2e+ 7h and the latter form is "simpler".

(There may be example where which form is "simpler" than the other is in the eye of the beholder!)

 

[JeffM]

To avoid repetition, just assume that I have said what others have already said.

(1) We change the form of an expression to make it easier to work with or to understand. (See # 6 for elaboration.)

(2) Changing the form of an expression to make it easier to work with frequently but not always involves simplifying it. Changing the form to make it more understandable almost always involves simplifying it.

(3) Changing the form of an expression DOES NOT CHANGE its numeric value. It is important to understand this because when we get to changing an equation, which involves two expressions, we almost always change the numeric value of both expressions by an equal numeric amount.

(4) To simplify an expression, we use with very high frequency four of the field properties, which are generalizations of arithmetic facts.
(i) a + b = b + a and (a + b) + c = a + (b + c), meaning that changing the order in which we do addition does not change numeric value.
(ii) a * b = b * a and (a * b) * c = a * (b * c), meaning that changing the order in which we do multiplication does not change numeric value.
(iii) a * (b + c) = (a * b) + (a * c).
(iv) a - b - c - d = a - (b + c + d), meaning subtracting several numbers in succession gives the same value as subtracting the sum of those numbers.

(5) One of the simplest ways to simplify an expression is to gather like terms and reduce, which means to bring together all terms with the identical variable and sign, summarize the terms with like variable and sign, and net terms with the same variable.

Consider the following messy expression.

[MATH]4x - 9x + 2y + 3z - 13x + 4z + 2x - 9y - x - 5z + 17x[/MATH].

To simplify it, first gather like terms.

[MATH]4x - 9x + 2y + 3z - 13x + 4z + 2x - 9y - x - 5z + 17x =[/MATH]
[MATH](4x + 2x + 17x) - (9x + 13x + x) + 2y - 9y + (3z + 4z) - 5z =[/MATH]
[MATH]x(4 + 2 + 17) - x(9 + 13 + 1) + 2y - 9y + z(3 + 4) - 5z.[/MATH]
Next, summarize terms with a like variable and sign.

[MATH]x(4 + 2 + 17) - x(9 + 13 + 1) + 2y - 9y + z(3 + 4) - 5z =[/MATH]
[MATH]23x - 23x + 2y - 9y + 7z - 5z.[/MATH]
Finally, net terms with the same variable.

[MATH]23x - 23x + 2y - 9y + 7z - 5z = -7y + 2z = 2z - 7y.[/MATH]
Which is easier to understand, 4x - 9x + 2y + 3z - 13x + 4z + 2x - 9y - x - 5z + 17x or 2z - 7y?

The procedure is actually quite routine. Later you will learn to use shortcuts to make this routine quicker, but this is what simplifying expressions is basically about.