it seems so easy when you see the solution, but not so easy when your making holes in the wall with your forehead.(Coffee)

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

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

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

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