$\displaystyle 78 - 6 (x - 8)$

$\displaystyle = 78 - 6 \cdot x - (- 6) \cdot 8$

$\displaystyle = 78 - 6x + 48$

$\displaystyle = - 6x + 78 + 48$

$\displaystyle = - 6x + 126$

In order to solve for $\displaystyle x$, move $\displaystyle x$ to one side of the equation and everything else to the other. To do this, subtract $\displaystyle 4$ from both sides.

The first step is to apply the distributive property. Don't forget to distribute the negative in the second parenthesis!

$\displaystyle 5(b-3)-3(2b+4)$

$\displaystyle (5\times b)-(5\times3)+(-3\times2b)+(-3\times4)$

$\displaystyle 5b-15-6b-12$

Next, combine the variables and the numbers. This gives us:

$\displaystyle 5b-6b-15-12$

$\displaystyle -b-27$

$\displaystyle \frac{1}{5} - 6x$ is one fifth decreased by $\displaystyle 6x$. $\displaystyle 6x$ is the product of six and a number.

Consequently, $\displaystyle \frac{1}{5} - 6x$ is "one fifth decreased by the product of six and a number".

$\displaystyle \frac{x-9}{13}$ is thirteen divided into $\displaystyle x - 9$.

$\displaystyle x - 9$ is the difference of a number and nine.

Therefore, 

$\displaystyle \frac{x-9}{13}$ is "thirteen divided into the difference of a number and nine".

$\displaystyle 14x - 5 (x + 8)$

$\displaystyle = 14x - 5 \cdot x - 5 \cdot 8$

$\displaystyle = 14x - 5x - 40$

$\displaystyle = (14- 5) x - 40$

$\displaystyle = 9x - 40$

$\displaystyle 8 (x - 7) - 3(x + 2)$

$\displaystyle = 8 \cdot x -8 \cdot 7 - 3 \cdot x + (-3) \cdot 2$

$\displaystyle = 8x -56 - 3 x -6$

$\displaystyle = 8x - 3 x -56 -6$

$\displaystyle =( 8 - 3 ) x - (56 + 6)$

$\displaystyle =5 x - 62$

This problem is just a matter of grouping together like terms.  Remember that terms like $\displaystyle xy$ are treated as though they were their own, different variable:

$\displaystyle 3x + 4x - 3y - 15y + 2xy$

The only part that might be a little hard is:

$\displaystyle -3y - 15y$

If you are confused, think of your number line.  This is like "going back" (more negative) from 15.  Therefore, you ranswer will be:

$\displaystyle 7x + 2xy - 18y$

This problem really is a trick question.  There are no common terms among any of the parts of the expression to be simplified.  In each case, you have an independent variable or set of variables: $\displaystyle x, y, xy,$ and $\displaystyle yz$.  Therefore, do not combine any of the elements!

Remember, when there is a subtraction outside of a group, you should add the opposite of each member.  That is:

$\displaystyle 5x + 3y - (4x + 2y) = 5x + 3y + (-4x) + (-2y)$

That is a bit confusing, so let's simplify.  When you add a negative, you subtract:

$\displaystyle 5x + 3y - 4x - 2y$

Now, group your like variables:

$\displaystyle 5x- 4x + 3y - 2y$

Finally, perform the subtractions and get: $\displaystyle x + y$