To simplify this problem we need to combine like terms.

$\displaystyle 6x-7y+3y-8x$

$\displaystyle =6x-8x-7y+3y$

$\displaystyle =-2x-4y$

To simplify this problem we need to combine like terms.

$\displaystyle -3y-5x+6x+4y$

$\displaystyle =-5x+6x-3y+4y$

$\displaystyle =x+y$

Simplify the following expression:

$\displaystyle 14y-12y-2y$

Let's begin by subtracting the 12y

$\displaystyle 14y-12y-2y=2y-2y$

From here, our answer should be apparent:

$\displaystyle 2y-2y=0$

So our answer is just 0

Simplify the following expression:

$\displaystyle 5q^7-3q^7-6q^6$

We can only subtract variables with the same exponent. 

In this case, we can only combine the first two terms.

To do so, keep the exponents the same and subtract the coefficients.

$\displaystyle (5-3)q^7-6q^6=2q^7-6q^6$

So our answer is:

$\displaystyle 2q^7-6q^6$

Simplify the following expression:

$\displaystyle 6x^7-5x^7-185x^8$

Now, to complete this, we need to realize that we can only subtract variables with the same exponent.

In this case, we can only combine our first two terms, because they both have an exponent of 7. The third term has an exponent of 8, so it cannot be combined and must be left as is.

$\displaystyle 6x^7-5x^7-185x^8=(6-5)x^7-185x^8=x^7-185x^8$

So, our answer must be:

$\displaystyle x^7-185x^8$

$\displaystyle 5x + 7y - 8x + 9y$

$\displaystyle =5x - 8x + 7y + 9y$

$\displaystyle =\left (5 - 8 \right ) x + \left (7 + 9\right ) y$

$\displaystyle =-3x + 16y$

$\displaystyle 10x-14y+7x-3y+15$

First, rewrite the problem so that like terms are next to each other.

$\displaystyle =10x+7x-14y-3y+15$

$\displaystyle =(10x+7x)+(-14y-3y)+15$

Next, evaluate the terms in parentheses.

$\displaystyle (10x+7x)=17x$

$\displaystyle (-14y-3y)=-17y$

Rewrite the expression in simplest form.

$\displaystyle =17x-17y+15$

$\displaystyle 15x^{2}+3x-7x^{2}+4x-5$

First we rewrite the problem so that like terms are together.

$\displaystyle =15x^{2}-7x^{2}+3x+4x-5$

Next we can place like terms in parentheses and evaluate the parentheses.

$\displaystyle =(15x^{2}-7x^{2})+(3x+4x)-5$

$\displaystyle 15x^{2}-7x^{2}=8x^{2}$

$\displaystyle 3x+4x=7x$

Now we rewrite the equation in simplest form.

$\displaystyle =8x^{2}+7x-5$

The first step in simplifying this expression is to get the binomial out of the parentheses. It's important to note you cannot further simplify this binomial first, since there are no like terms in it.

Since you have a minus sign in front of the binomial, you need to flip the sign of both terms inside the parentheses to get rid of the parentheses (similar to distributing a negative one across the binomial):

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

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

Now you are able to combine like terms, making sure that exponents on the variables match exactly before you combine. The first and fourth terms are like terms, and the second and third terms are like terms.

To combine those terms, keep the variables and exponents the same and add up the coefficients. The first term has a coefficient of $\displaystyle 1$ and the fourth term has a coefficient of $\displaystyle 2$, so they add up to a total of $\displaystyle 3$. The second term has a coefficient of $\displaystyle 4$ and the third term has a coefficient of $\displaystyle -2$, so they add up to a total of $\displaystyle 2$.

This brings you to the final, simplified answer:

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

$\displaystyle 3x^2y^3 + 2x^3y^2$

Combine like terms:
$\displaystyle x^4+2x+3x^2-4x+7x^2=x^4+(3x^2+7x^2)+(2x-4x)$

$\displaystyle =x^4+10x^2-2x$