How do you do this?

solve for x:

ax = bx - c

Thank you.

-r

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- May 22nd 2010, 08:40 AMirichi09solve this equation for x
How do you do this?

solve for x:

ax = bx - c

Thank you.

-r - May 22nd 2010, 08:52 AMAnonymous1
$\displaystyle ax = bx - c$

$\displaystyle \implies ax- bx = -c$

$\displaystyle \implies x(a- b) = -c$

$\displaystyle \implies \boxed{x=-\frac{c}{a-b}}$ - May 22nd 2010, 09:12 AMfreetibet
solve for x:

ax = bx - c

solution: ax - bx = c

(a -b)x = c

x = c/(a - b) - May 22nd 2010, 10:07 AMtreborleinadcorrect solution
This is the correct solution:

$\displaystyle

ax = bx - c

$

$\displaystyle

\implies ax- bx =-c

$

$\displaystyle

\implies x(a- b) =-c

$

$\displaystyle

\implies \boxed{x=-\frac{c}{a-b}}

$ - May 22nd 2010, 03:46 PMirichi09thank you for helping
it seems so easy when you see the solution, but not so easy when your making holes in the wall with your forehead.(Coffee)

- May 22nd 2010, 04:38 PMArchie Meade
To protect the wall from your forehead, you need to know how to think it through.

Solve for x means we are being asked to write x=?

What we are given contains x in 2 places, on either side of the equal sign.

When both sides are equal we can perform the exact same operation to both sides.

For instance 5=5

If we subtract 3 from both sides we'll have 5-3=5-3 which is 2=2

The sides may not be what they were originally but they are still equal.

Hence

$\displaystyle ax=bx-c$

has x on both sides.

To get the x's on the same side, subtract bx from both sides,

the two sides will still be equal...

$\displaystyle ax-bx=bx-bx-c$

$\displaystyle bx-bx=0$

therefore we have

$\displaystyle ax-bx=-c$

If you have 5 boxes of apples, 20 apples per box

and you give 3 boxes to your sister, how many apples will you have ?

$\displaystyle 5(20)-3(20)=(5-3)(20)$

You can calculate 100-60=40 or 2(20)

You get the same answer.

Hence we get x in one place by factorising as above.

$\displaystyle ax-bx=-c$

$\displaystyle x(a-b)=-c$

This is "x multiplied by both a and -b".

To move the (a-b) away from the x, to leave x standing alone,

we divide both sides by (a-b), just as

$\displaystyle 3x=6\ \Rightarrow\ \frac{3x}{3}=\frac{6}{3}$

$\displaystyle \frac{3}{3}x=\frac{6}{3}$

As $\displaystyle \frac{3}{3}=1$

this is

$\displaystyle x=2$

Hence

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

$\displaystyle \frac{(a-b)}{(a-b)}=1$

$\displaystyle x=-\frac{c}{a-b}=\frac{c}{b-a}$ - May 22nd 2010, 09:26 PMbrumby_3
Nice explanation Archie!