The first step is to translate the words, "if $\displaystyle 15$ is added to $\displaystyle \frac{1}{3}$ of another number, the result is $\displaystyle 24$," into an equation. This gives us:

$\displaystyle 15+\frac{1}{3}x=24$

Subtract $\displaystyle 15$ from each side. 

$\displaystyle \frac{1}{3}x=9$

Multiply each side by $\displaystyle 3$. 

$\displaystyle x=27$

Therefore, $\displaystyle 27$ is the correct answer. 

When adding and subtracting variable, you can only combine like variables.  

That means all of the $\displaystyle x$ variables are solved separately from the $\displaystyle y$ variables.  

Then you just add and subtract the constants normally so $\displaystyle 5x-2x=3x$ and $\displaystyle 7y+3y=10y$.  

So the final answer is $\displaystyle 3x+10y$.

When adding variables together, you must first make sure you are combining the same variable.  So, in this case

$\displaystyle a+4a$

we can see that both terms contain the variable a.  Therefore, we can combine them. 

Now, when we combine them, we can think of the variables as objects.  So, we can say were are combining an apple and 4 apples together.  So,

$\displaystyle \text{apple} + 4\text{ apples } = 5\text{ apples}$

We can simplify our problem the same way.

$\displaystyle a +4a = 5a$

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

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

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

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

$\displaystyle = 9x - 40$

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

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

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

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

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

$\displaystyle =5 x - 62$

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:

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

You need to begin by distributing the minus sign through the whole group $\displaystyle (13b - 2a)$.  This gives you:

$\displaystyle 15a + 23b - (13b - 2a) = 15a + 23b - 13b - (-2a)$

Simplifying the double negative, you get:

$\displaystyle 15a + 23b - 13b + 2a$

Now, you can move the like terms next to each other:

$\displaystyle 15a + 2a + 23b - 13b$

Finally, simplify:

$\displaystyle 17a + 10b$

Begin by distributing the $\displaystyle -3$:

$\displaystyle 57x - 3(4x) - (-3)(5y)$

Multiply each factor:

$\displaystyle 57x - 12x - (-15y)$

Change the double negation to addition:

$\displaystyle 57x - 12x +15y$

Combine like terms:

$\displaystyle 45x + 15y$

Begin by distributing the $\displaystyle -2$:

$\displaystyle 55x - 13xy + (-2)5y + (-2)10xy$

Multiply all factors:

$\displaystyle 55x - 13xy + (-10)y + (-20)xy$

Group together the only like factor ($\displaystyle xy$):

$\displaystyle 55x - 13xy-20xy -10y$

Combine like terms:

$\displaystyle 55x - 33xy -10y$

$\displaystyle 8(x+7) - 3 (x + 10)$

$\displaystyle = 8 \cdot x+ 8 \cdot 7 - 3 \cdot x - 3 \cdot 10$

$\displaystyle = 8 x+ 56 - 3x - 30$

$\displaystyle = 5x+ 26$