Trying to teach myself equations

[Vincent Vega]

I'm trying to teach myself equations and I'm stuck on equations with just letters. I can easily solve for x with: $$34 - 10x = 6x + 2$$

x=2

I'm having trouble with:
$$ax = bx + c$$

 

[Integral]

How did you arrive at the conclusion x=2 in your example?

 

[Vincent Vega]

I subtracted 2 from each side using the "golden rule of equations"

Whatever you do to one side of an equation, do exactly the same thing to the other side.

.

That gave $$32 - 10x = 6x$$

then $$32 - 10x + 10x =6x + 10x$$

This LaTeX is giving me some trouble here, bare with me.
thanks for your help!

 

[d_leet]

I subtracted 2 from each side using the "golden rule of equations" .

I think you have the concept down perfectly. Now can you apply what you did with that equation to the other one

ax = bx + c ?

 

[Vincent Vega]

then $$\frac{32} {16}$$ $$\frac{16x} {16}$$ x=2

 

[chroot]

Move ax and bx to the same side of the equation by subtracting bx from both sides:

ax - bx = c

Factor out the x on the left:

(a - b) x = c

Divide both sides by (a-b):

x = c / (a-b)

You already know how to do algebra! The fact that they're letters instead of numbers changes nothing.

- Warren

 

[konartist]

I know where you're coming from though with the letters and all. However, in pre calc we use a lot of problems with letters instead of numbers to find other formulas.

 

[Vincent Vega]

Thanks for the help!

When chroot factored out the $$x$$ on the left, what rule/ property/operations did he use to achieve that (and when to use parenthesis ( ).. )? My book never showed me examples or really explained why or how to do that.(atleast from what I read)

All I know is:
Postualates of equality/operations


Also, I know there can be several ways to work out an equation, but where do you people usually start to break down these problems when solving for x.

I usually do it the wrong way before I get it the right way. I know there are no shortcuts to learning but I'm just wondering if I overlooked something here.

 

[chroot]

I just used the distributive property of multiplication.

(a-b)*x = ax - bx.

- Warren

 

[Vincent Vega]

distributive property of multiplication. Ha, thanks

 

[chroot]

Got it?

- Warren

 

[Vincent Vega]

Got it?

- Warren

I need to search around for more practice equations until it comes natural to me.

My question is what does the / stand for in the answer ... divide?

 

[Vincent Vega]

b + ax = c
b - b + ax = c - b
ax = (c - b)

What are we doing to both sides to isolate the x on that last line? I got the answer only because I saw a pattern, but I'm unsure how I got it. What let me isolate x by moving a to the other side of the problem?

x = (c - b)/a

 

[Hootenanny]

b + ax = c
b - b + ax = c - b
ax = (c - b)

What are we doing to both sides to isolate the x on that last line? I got the answer only because I saw a pattern, but I'm unsure how I got it. What let me isolate x by moving a to the other side of the problem?

x = (c - b)/a

Think of it like this; You divided both sides by $a$. You could add an extra step between the penultimate and final line if you like;

$$ax = (c-b)$$

$$\frac{ax}{a} = \frac{(c-b)}{a} \Rightarrow \frac{\not{a}x}{\not{a}} = \frac{(c-b)}{a}$$

$$x = \frac{(c-b)}{a}$$

You can do this because you did the same thing to both sides, i.e. divided both sides by a. You can do anything to can equation as long as you do it to both sides of the equals sign.

 

[Vincent Vega]

Yes , thanks for showing me that, I understand them now.
However, when I look at this problem I want to try using the d first on both sides... Is there something you see in the equation that tells you to use b first. Because while using d I came to this

ax - b = cx + d

ax- b - d = cx + d - d

ax - (b - d) = cx

ax - ax - (b - d) = cx - ax


$$\frac{-(b - d)}{c - a}$$$$\frac{cx - ax}{c - a}$$

-(b - d)/(c-a) = x .... what does the -(b - d) part equal? Is it (b + d)?

 

[Vincent Vega]

Then I tried the b first:

ax - b = cx + d

ax - b + b = cx + d + b

ax = cx + (d + b)

ax - cx = cx - cx + (d + b)

$$\frac{ax - cx}{a - c}$$$$\frac{(d+b)}{a-c}$$

x= (d+b)/(a-c)

 

[Libertine]

ax- b - d = cx + d - d

ax - (b - d) = cx

That step is incorrect:
-b-d =/= -(b-d)

 

[Hootenanny]

A simpler way of doing to would be to move the cx over to the left first. Then the b over to the right. Then factorise;

$$ax - b = cx + d$$

Add b to both sides and subtract cx from both sides

$$ax - cx = d + b$$

Factorise x;

$$x(a - c) = d + b$$

Divide both sides by the bracket;

$$x = \frac{d+b}{a-c}$$

To answer your earlier question;

what does the -(b - d) part equal? Is it (b + d)?

No $-(b-d) = d-b$. You have made an error in your first attempt.

This step

$$\frac{ax - cx}{a - c}= \frac{(d+b)}{a-c}$$
$$x= \frac{(d+b)}{a-c}$$

Is not correct because

$$\frac{ax - cx}{a - c} \not{=} x$$

[Edit] Libertine got there before me [/Edit]

 

[Vincent Vega]

Ok, I've been doing them fairly good now using th distributive property of multiplication chroot showed me. I did these correct I presume?

10 + 5x = 100 - 4x

(4 + 5)x = 100 - 10

$$\frac{9x}{9}= \frac{90}{9}$$

x=10


--------------------------------------
10 + 5x = 100 - 4x

10 - 10 + 5x = 100 - 10 - 4x

5x = 90 - 4x

5x + 4x = 90

$$\frac{9x}{9}= \frac{90}{9}$$

x=10


Thanks for taking the time with me.


cheers everyone

 

[chroot]

Looks like you've got it, Vincent Vega.

I assume you understand the objections raised by Hootenanny and Libertine in the last few posts?

- Warren

 

[Vincent Vega]

Looks like you've got it, Vincent Vega.

I assume you understand the objections raised by Hootenanny and Libertine in the last few posts?

- Warren

Yes. Much love guys. I'm on negative numbers and integers which seems to be flowing now, hopefully.




________
"Never be first; try to be second" __ Enrico Fermi