Help with equations

[vashkey]

So I'm getting ready for the ACT and I'm running through math again and I got a book to help go through it but it's not especially descript at times.

I'm at "equations" and I seem to get how it works most of the time but some of the warm up problems have me a little confused.

For example...

5x - 6 = 2x + 9.


In the first step it tells you to to subtract 2x from both sides. Simple enough, but I'd just like to know, why not 5x instead. I'm not saying it's wrong but understanding would help alot. Like wise, the next step is to add 6 to both sides, but why not subtract 9 from both instead?


Also, theres this problem I ran into...

2x + 2(3x + 2) - 9 = (3x - 9) + 3

In the first step it shows that you need to multiply the number on the outside of the first set of parentheses with the numbers inside so that the problem now looks like this

2x + 6x + 4 - 9 = 3x - 9 + 3

Makes sense so to me far...

Then you need to add/subtract everything on each side of the equal sign. After that step this is what the equation looks like

8x - 5 = 3x - 6

Still makes sense.


Then subtract 3x from both sides to get....


5x - 5 = -6

then add five to both sides then you get...

5x = -1


At this point you're supposed to divide 5 from both sides... But the book leaves it at this

x= -1 and then says to the side that the solution set is -1/5. If thats the case then why isn't that what x is equal to?

 

[vashkey]

Oh, alright. Thank you.

Here's another question I'm confused about though.

3(m - 4 ) - (4m - 11) = -5

In the next set it becomes...

3m - 12 -4m + 11 = -5

Why was - 11 changed to + 11?


In the next step it becomes...

-m - 1 = -5

Which makes sense ignoring my confusion about the 11 becoming positive for the moment.

Then it becomes

-m = -4

Then in the final step it's the same thing except m is no longer negative. Why?

 

[vashkey]

I think I get it. Since -4 equals -m then there's no point in expressing m with a negative?

 

[vashkey]

Oh... Wait. I totally missed that the -4 also became a positive 4. This makes sense. I'm sorry and thank you very much for the assistance.

 

[vashkey]

-(-5) = 5
Means remove a previous -5 operation or minus off a -5 operation;
so if you do that, you're increasing, right? Hope I didn't confuse you :cool:
10 - 5 = 5
5 - (-5) = 10
Like, you changed your mind and wanted to cancel the first operation...

Yeah, I got that from the other explanation. I just hadn't thought of how it might apply to the rest of the number.

Like with 3(m - 4), I know t multiply both m and - 4 by 3. I just did know having a - outside a pair of parentheses would effect more than just the first number in a set of parentheses, till now.

 

[vashkey]

Heres another one I'm confused about. Forgive me, I'm not sure if this is the correct way to express the problem with a keybaord.

2x + 3 + 1 = 2x - 1
3 4 6


The first step is to multiply all the numerators by what ever it takes to make their corresponding denominators to become 12. This makes sense since all denominators need to be the same when adding or subtracting fractions. However, what confuses me is that in the book for the rest of the equation it appears to ignore the denominators.

So for step one they make it look like this...

8x + 4 + 3 = 4x - 2

Then

8x + 7 = 4x - 2

then

4x + 7 = -2

then

4x = -9


I understand what they're doing... but then comes time to divide by 4 and we get...

x = 9
4

And next to it it says -9
2

is the solution set.

Why? What happened to 12 being the denominator? Even then why does nine become negative and 4 become 2?

 

[vashkey]

In the first equation 3 should be the denominator of 2x + 3, 4 should be the denominator of 1 and 6 should be the denominator of 2x - 1.

Toward the end of my post 4 should be under 9 and 2 should be under -9.

I spaced em out like that in my post but once it posts those numbers reverted to the beginning of their respective lines. I don't know how to properly type out those fractions.

 

[mmm4444bot]

If you want to "draw" fractions on this board, you need to enclose your "drawing" within the [CODE] and [/CODE] tags. Otherwise, the system will strip out your extra spacings.

Also, switch the font in that section to Courier New so that stuff lines up properly while you type. Use the [Preview Post] button to proofread your "drawing", before clicking [Submit Reply].

As an alternative to "drawing" fractions, it is easier to simply use grouping symbols around numerators/denominators that contain more than one term. We text that equation like so:

(2x + 3)/3 + 1/4 = (2x - 1)/6



By the way, the solution to this equation is not x = -9/2.

The author made yet more arithmetic mistakes, and the publisher's proofreader was out to lunch.

Here's one mistake.

4(2x + 3) is 8x + 12, not 8x + 4.



It seems to me that you got your hands on a very crappy book. Is it one of the "Math for Dummies" books? I proofread a paperback in that series, and it contained more mistakes and misstatements than it did pages. (The title of that series probably refers to the publisher. :roll

The first step is to multiply all the numerators by what ever it takes to make their corresponding denominators to become 12

This is part of a horrible explanation.

12 is the LCD (lowest common denominator).

We multiply each fraction by the LCD, so that each denominator cancels.

Doing that yields:

4(2x + 3) + 3 = 2(2x - 1)

Go from there, and see whether you can discover the true solution.

Note: You can always confirm a solution candidate by (1) substituting that value for the variable in the original equation, (2) doing the subsequent arithmetic, and (3) verifying that the end result is a true equation.

If the author were to have checked this way (assuming requisite skills), the result would have been -7/4 = -5/3 (not a true equation, so x=-9/2 is clearly incorrect).



PS: In the future, please start a new thread for each new exercise that you wish to discuss.

And thank you so much for being specific, in your posts. Most people seeking help here don't say diddly. Cheers :cool:

 

[Deleted member 4993]

Oh, alright. Thank you.

Here's another question I'm confused about though.

3(m - 4 ) - (4m - 11) = -5

In the next set it becomes...

3m - 12 -4m + 11 = -5

Why was - 11 changed to + 11?


In the next step it becomes...

-m - 1 = -5

Which makes sense ignoring my confusion about the 11 becoming positive for the moment.

Then it becomes

-m = -4

Then in the final step it's the same thing except m is no longer negative. Why?

I think the final answer given in the book is:

m = 4

This you get by multiplying bothe sides by "-1".

Now you are establishing the value of "m" not "-m".

Similarly, you won't leave the answer as "2m = 8" - you'll divide both sides by "2" to reduce the answer to "m = 4".