order of operations?

[e1e1e1e1]

Is there a certain order of operations I need to follow when doing a problem like this?

1/5n + 2 = 3.2

I tried first multiplying 1/5n and 3.2 both by 5 to get:

n+2=16

n=14

But the answer was wrong. Turns out I was supposed to deal with the +2 first (subtract 2 on both sides) to get:

1/5n = 1.2

THEN do the multiplication by 5 to get

n=6

So clearly the order of operations matters, but how do I know what to do first?

(PEDMAS obviosly wouldn't work because that's what I tried first and got the wrong answer)

 

[fcabanski]

The thing to do in an algebra problem is to isolate the variable. You don't have to get all the terms that don't contain n on one side of the equation first, but that makes the work simpler.

For example, 5x +2 = -13. You can divide every term by 5: x+2/5 = -13/5.

Or you can first subtract 2 from both sides: 5x=-15. This is simpler. Now divide each side by 5.

1/5n + 2 = 3.2: the first term is divided by 5n. Multiplying all terms by 5 removes the 5 from the denominator. But what about the n?

Also, it's simpler if you first subtract 2 from both sides.

 

[Quaid]

The Order of Operations (PEMDAS) is used for evaluating expressions.

This is not the same as the order in which we carry out algebraic steps while solving an equation.

For example, when evaluating, we do multiplication before addition.

5 * 10 + 2 = ?

5 * 10 = 50

50 + 2 = 52

(We multiplied first, then added, according to PEMDAS)​

When solving a simple equation, we add stuff before multiplying stuff.

5x + 2 = 0

Add -2 to each side: 5x + 2 - 2 = 0 - 2

5x = -2

Multiply each side by 1/5: (1/5)(5x) = (1/5)(-2)

x = -2/5

(We added first, then multiplied)​

​

So, we see that the order is different, depending on the task.

Cheers

 

[HallsofIvy]

Is there a certain order of operations I need to follow when doing a problem like this?

1/5n + 2 = 3.2

This is extremely ambiguous. It could be read as (1/5)n+ 2= 3.2 or 1/(5n)+ 2= 3.2 or 1/(5n+2)= 3.2.
Use parentheses!

If it is (1/5)n+ 2= 3.2 then, yes, it helps to know the "order of operations". Since multiplication has higher priority than addition, if you were evaluating this for a given value of n, you would first multiply 1/5 and n, then add 2.

But to solve for n, you do the opposite operations in the opposite order. That is, first subtract 2 from both sides then divide both sides by 1/5 (which is the same as multiplying by 5).

I tried first multiplying 1/5n and 3.2 both by 5 to get:

n+2=16

n=14

But the answer was wrong. Turns out I was supposed to deal with the +2 first (subtract 2 on both sides) to get:

1/5n = 1.2

THEN do the multiplication by 5 to get

n=6

So clearly the order of operations matters, but how do I know what to do first?

(PEDMAS obviosly wouldn't work because that's what I tried first and got the wrong answer)

 

[e1e1e1e1]

The Order of Operations (PEMDAS) is used for evaluating expressions.

This is not the same as the order in which we carry out algebraic steps while solving an equation.

[...] when evaluating, we do multiplication before addition.

When solving [...] we add stuff before multiplying stuff.

[...] the order is different, depending on the task.

Thanks. I think this makes sense.

It is this algebriac "sequence of steps" (to avoid confusion with "Order of Operations") that I am asking about. And, if I am correctly understanding what has been explained in this thread, (unlike PEDMAS) you do the addition and/or subtracion steps (if applicaple) first, then do the multiplication and/or division steps (if applicable).

 

[e1e1e1e1]

Your error (did not multiply 2 by 5) has NOTHING to do with order of operations.
Example:
10/5 + 3 = 5

(10/5)*5 + 3 = 5*5
10 + 3 = 25 .... wrong!

(10/5)*5 + 3*5 = 5*5
10 + 15 = 25 .... correct!

Each term must be multiplied...that's quite basic...

Thanks. I totally missed that! I needed to multiply everything on the left side of the equal sign by 5 and failed to do so.

Of course, had I done the addition and/or subtraction steps first (as apparently I was supposed to), I would have avoided this. But then I would not have understood the fundamental error you pointed out.

 

[e1e1e1e1]

This is extremely ambiguous. It could be read as (1/5)n+ 2= 3.2 or 1/(5n)+ 2= 3.2 or 1/(5n+2)= 3.2.
Use parentheses!

It can be confusing sometimes. Let me give another example on a Khan Academy video:

He writes:
3(9-2y)+5y=20

27-6y+5y=20

27-y=20

But wouldn't it be clearer if the second step was written as:
27 + -6y + 5y = 20

Otherwise I might get:
27-11y=20

What is the correct way to do this?

 

[Deleted member 4993]

It can be confusing sometimes. Let me give another example on a Khan Academy video:

He writes:
3(9-2y)+5y=20

27-6y+5y=20

27-y=20

But wouldn't it be clearer if the second step was written as:
27 + -6y + 5y = 20

Otherwise I might get:
27-11y=20

What is the correct way to do this?

Khan_academy way is the correct way - because -

3 * (-2) = -6

I generally don't get into supporting Khan_academy - people may accuse me of "nepotism"!!

 

[e1e1e1e1]

Khan_academy way is the correct way - because -

3 * (-2) = -6

Yes, but supposing you didn't see that first step...
suppossing you only had in front of you:
27-6y+5y=20

it isn't clear to me whether that means:
27 + -6y + 5y = 20

or:
27 - 6y + 5y = 20

 

[Quaid]

27 + -6y + 5y = 20

or:

27 - 6y + 5y = 20

The equations above are equivalent.

Here's another way to view subtraction: in terms of addition!

27 - 6y

is the same as

27 + (-1)(6y)

In other words, subtracting 6y from 27 is the same as adding the opposite of 6y to 27.

Here's a more symbolic example. Subtract B from A. That's the same as adding the opposite of B to A.

A - B = A + (-1)(B) = A + -B

Now, when we write expressions like 27 + -6 or A + -B, those negative signs are not subtraction symbols. Those negative signs are an abbreviation for "multiply by -1".

We generally see A - B and then think A + -B. That is, we come to view subtraction as adding the opposite; we don't usually write it down.




Here's my thought process, when I look at 27 - 6y + 5y = 20

1. -6y and 5y are like-terms; combine them

2. (-6 + 5)*y is (-1)*y

3. Adding -y to 27 is the same as 27 - y

4. write 27 - y = 20 and continue...

 

[Quaid]

Another example.

A - (A + B) = ?

We are subtracting the sum (A+B) from A.

That's the same as adding the opposite of (A+B) to A.

-(A + B) + A

Again, that negative sign in front of the parentheses means "multiply by -1".

(-1)(A + B) + A

Think: Distributive Property, to carry out the multiplication.

(-1)(A) + (-1)(B)​ + A

-A - B + A

And, if you want to think of that as -A + -B + A, that's great!

Now recognize the like-terms. We are combining positive A with negative A. See that?

-A + A results in zero, so we're left with just -B

Therefore, A - (A + B) = -B

With sufficient practice, your mind will fly through these steps in very short order. You will see A-(A + B) and think similar to, "I'm subtracting both A and B. A minus A is zero. And, zero minus B is -B."

:cool:

 

[e1e1e1e1]

Here's my thought process, when I look at 27 - 6y + 5y = 20

1. -6y and 5y are like-terms; combine them

when I look at
27 - 6y + 5y

I see like terms of positive 6y and positive 5y; combine them
27 - 11y

But if it were writen as
27 + -6y + 5y

then I would understand that we are combining like terms of -6y and 5y to get
27 + -1y

what am i doing wrong?

 

[HallsofIvy]

when I look at
27 - 6y + 5y

I see like terms of positive 6y and positive 5y; combine them
27 - 11y

But if it were writen as
27 + -6y + 5y

then I would understand that we are combining like terms of -6y and 5y to get
27 + -1y

what am i doing wrong?

You are mistaking "-6y+ 5y" for "-(6y+ 5y)". By the "order of operations" multiplication has higher priority than addition:-6y and 5y before adding.

 

[e1e1e1e1]

You are mistaking "-6y+ 5y" for "-(6y+ 5y)". By the "order of operations" multiplication has higher priority than addition:-6y and 5y before adding.

Where is there multiplication in this problem?

27 - 6y + 5y

I don't see any multiplication to be done. Just addition and subtraction.

 

[Quaid]

when I look at
27 - 6y + 5y

I see like terms of positive 6y and positive 5y

In algebra, that is an incorrect interpretation. Subtracting the term makes it negative.

if it were writen as
27 + -6y + 5y

then I would understand

You won't see the addition of a negative term written like that very often (we think it, but we don't write it that way). We show +-6y by subtraction, instead.

Likewise, we don't write 27 - +6y + 5y, either.


27 + -6y + 5y means 27 - 6y + 5y and vice versa.

With more exposure, you will come to recognize subtraction as adding a negative quantity to something.


If it helps to interpret an expression, you can always use the Commutative Property, to reorder terms.

27 + 5y - 6y

Now what do you think? I just swapped the positions of the 5y and -6y terms. I can do that, without changing the value of the overall expression, because terms can be combined in any order. That's what the Commutative Property tells us.

I could also write 27 - 6y + 5y as follows; all of the versions are equivalent.

5y + 27 - 6y

5y - 6y + 27

-6y + 27 + 5y

-6y + 5y + 27

How about this last one?

When you look at it, do you still see a positive 6y being added to 5y?

Cheers :cool:

 

[e1e1e1e1]

In algebra, that is an incorrect interpretation. Subtracting the term makes it negative.




You won't see the addition of a negative term written like that very often (we think it, but we don't write it that way). We show +-6y by subtraction, instead.

Likewise, we don't write 27 - +6y + 5y, either.


27 + -6y + 5y means 27 - 6y + 5y and vice versa.

With more exposure, you will come to recognize subtraction as adding a negative quantity to something.


If it helps to interpret an expression, you can always use the Commutative Property, to reorder terms.

27 + 5y - 6y

Now what do you think? I just swapped the positions of the 5y and -6y terms. I can do that, without changing the value of the overall expression, because terms can be combined in any order. That's what the Commutative Property tells us.

I could also write 27 - 6y + 5y as follows; all of the versions are equivalent.

5y + 27 - 6y

5y - 6y + 27

-6y + 27 + 5y

-6y + 5y + 27

How about this last one?

When you look at it, do you still see a positive 6y being added to 5y?

Cheers :cool:

Thanks, this is all actually very helpful.

Understanding the "conventions" of how things are written is helpful and makes things clearer.

The reordering examples (using the Commutative Property) are especially helpful in explaining the concepts. In each of your examples it is clear to me that it would work out to 27 - y and not 27 - 11y so that helps a lot.

I think as long as I read the expression...

27 - 6y + 5y

...from left to right FIRST, and think of it as something like...

"27 minus 6y plus 5y"
or
"27 take away 6y and add 5y"
or
"27 go left on the numberline a value of 6y then go right on the numberline a value of 5y"
etc...
I would ultimately get:
27 - y

I think my problem is I'm looking at:
27 - 6y + 5y

and I'm thinking "combine like terms" FIRST, and this has the effect of turning it into:
27 - (6y + 5y)

(sort of like what HallsofIvy is saying I'm doing)

So maybe what I need to do is FIRST read through the expression from left to right and understand that it is:
"27 minus 6y plus 5y"
then start looking for like terms and abstract out from that:
"minus 6y plus 5y"
and then do the combining of like terms (perhaps even reordering it as "5y minus 6y") to get:
"minus 1y" or "minus y" or "negative y"
to get:
27 - y