A very easy question about equations

Hello,

I understand you can do anything to one side of an equation as long as you do it to both sides. What I haven't picked up on is "where" you do it to the other side of the equation. For example, let's say you have...

8x + 10 = (3x - 10)/8 (just made that up)

You can remove the 10 from the left side as long as you add -10 to each side. So the left side becomes 8x =, but the right side you have to add -10. I am not sure what the rule is to decide "where" to add -10? I mean, do you remove it from the whole right side, so

8x = ((3x - 10)/8) - 10 ?

Like in this question/answer

Q: b + ax = 12 (solve for X)
A: x = (12 - b) / a

So to arrive at the answer they moved 'b' to the right side by removing it from 12. Makes sense, is the only thing on the right side in the first place. But the second step is to move 'a' to the right side, so they divide "all" of the right side by 'a'. What if when solving you did the solution in the other order, so you went

b + ax = 12
b + x = 12 / a <--- move 'a' to the right side by dividing both sides by it.
x = (12 / a) - b <--- move 'b' to the right side by subtracting it from both sides.

but "x = (12-b)/a" is a very different answer than "x=(12/a)-b"

for example, if x=2, b=2, a=5

the corrent answer figures up like this: 2 = (12-2)/5
my answer figures up like this: 2 = (12 / 5) - 2
which isnt even close to the same thing...

Where am I so confused?

If you following the guide of BOMDAS
Brackets
operators(ie powers)
multipy/division
addition/subtraction

This is how you solve a problem when reduction is required do them in order from start to finish(ie brackets first), but when moving items around you start from right to left.. ie do the addition/subtratcion first and work on.

This guide will always help you with the order.

I understand the order of operations when actually carrying out linear arithmetic, such as

6 x 5 + 1 - (2-4) = ? you would go

2-4, then 6 x 5, then + 1, then - 2.

What I am not understanding is the order of, and placement of terms when moving them from side to side in a linear equation. In my book, I've seen examples like

12x - 5 = 10 + 15

The first thing they do is the "addition"

12x - 5 = 25

The second thing they do is the "subtraction" to move the 5 over

12x = 30

The last thing they do is the "division", so

x = 30/12

So I am under the assumption that BOMDAS doesn't appear to be required when deciding which order to carry out the simplification of an equation, is that correct?

If not, then my confusion arises in even a simple example such as this...

Problem: b + ax = 12
Answer: x = (12 - b) / a

I am just not following the logic. I am assuming that first they did the "subtraction", to move b to the other side..

ax = 12 - b

then they did the "division" to drop the a from the left and move it to the right...

ax / a = (12 - b) / a, so... x = (12 - b) / a

So I can understand those steps, but they seemed to subtract first, then divide (against the standard order of operations). Which is fine, but then what prevents me from dividing first, if that is allowed that is where I get confused...

So I start with

b + ax = 12

then divide both sides by a to get rid of it from the left...

(b + ax) / a = 12 / a

so

b + x = 12 / a

now I want to move the 'b' to the right side, so I subtract it from each side...

x = (12 / a) - b .... my answer
x = (12 - b) / a .... the correct answer

They seem to evaluate to the same thing, so am I wrong? Or, maybe I am not wrong but not doing it an expected way? What is my mistake? I feel like Im misunderstanding some fundamental rule here.

when you have all the equation on one side
6 x 5 + 1 - (2-4) = ?
follow the BOMDAS rules from left to right to solve.

When rearranging equations (ie parts are on both sides) do the rearrangement using the reverse oreder of BOMDAS
12x - 5 = 10 + 15

b + ax = 12

as there are parts on both sides you need to rearrange using the reverse order of BOMDAS.

ax =12 - b
x= (12 - b)/a

does this help.

Yes, that helps a lot. I have gone back through my problems and gotten more correct now with that.

This one though,

Problem: ax = bx + c
Answer: x = c / (a - b)

I just do not see how they arrive at the answer at all.
The ax = bx part is confusing. Need to get x alone, so I think "divide both sides by 'a'...

x = (bx + c) / a

Hrm, now need to get x out of the right side... but that still leaves an X on both sides and is looking less like the answer. So I start over thinking maybe the put everything on the right side first to get the x's together...

ax = bx + c
ax - bx = c (subtract bx from each side)

...well now what? man I just dont get how they arrive at x = (c - b) / a

ax = bx + c


first need to gather the unknows (x ) together..

ax -bx = c

then take the common factor out...

x(a-b) =c

then move the (a-b) to the otherside

x= c / (a-b)

backToCollege said:

Problem: ax = bx + c
Answer: x = c / (a - b)
I just do not see how they arrive at the answer at all.
The ax = bx part is confusing. Need to get x alone, so I think "divide both sides by 'a'...
x = (bx + c) / a
Hrm, now need to get x out of the right side... but that still leaves an X on both sides and is looking less like the answer. So I start over thinking maybe the put everything on the right side first to get the x's together...
ax = bx + c
ax - bx = c (subtract bx from each side)
...well now what? man I just dont get how they arrive at x = (c - b) / a

Click to expand...

ax = bx + c
You're solving for x, so start by putting the x terms together:
ax - bx = c
Now you can factor this way:
x(a - b) = c
And now divide to end up:
x = c / (a - b)

Don't get too worried...it'll all come to you...

Ahhhh okay. Thank you very much. I see it now. So basically to get from

ax - bx
to
x(a - b)

Its using the distributive property of multiplication, but what was throwing me was the order. Silly me, If I'd have just reversed the letters to:

xa - xb, then the distributive examples ive seen would have kicked in...

x(a - b)

Thank you again, you're right it will all make sense soon, just takes practice.