Slope-intercept form is written as $\displaystyle \small y=mx+b$.

There is only one answer choice in this form:

$\displaystyle \small y=4x+2$

The slope-intercept form of a line is: $\displaystyle y=mx+b$, where $\displaystyle m$ is the slope and $\displaystyle b$ is the y intercept.

Below are the steps to get the equation $\displaystyle 2+4y=3x+10$ into slope-intercept form.

$\displaystyle 2+4y=3x+10 \textup{ (Subtract 2 from both sides)}$

$\displaystyle 4y=3x+8 \textup{ (Divide both sides by 4)}$ 

$\displaystyle y=\frac{3}{4}x+2$

First, we need to find the slope of the above line. 

Given two points, $\displaystyle (x_{1}, y_{1}), (x_{2}, y_{2})$, the slope can be calculated using the following formula:

$\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $\displaystyle x_{1}=-4, y_{1}=x_{2}= 0, y_{2}=8$:

$\displaystyle m = \frac{8-0}{0-(-4)} = \frac{8}{4} = 2$

Second, we note that the $\displaystyle y$-intercept is the point $\displaystyle (0,8)$. 

Therefore, in the slope-intercept form of a line, we can set $\displaystyle m = 2$ and $\displaystyle b = 8$:

$\displaystyle y = mx+b$

$\displaystyle y = 2x+8$

There are two ways that you can find the y-intercept for an equation.  You could substitute $\displaystyle 0$ in for $\displaystyle x$.  This would give you:

$\displaystyle 2y - 4*0 = 10*0 - 20$

Simplifying, you get:

$\displaystyle 2y = -20$

$\displaystyle y = -10$

However, another way to do this is by finding the slope-intercept form of the line.  You do this by solving for $\displaystyle y$:

$\displaystyle 2y = 14x - 20$

Just divide everything by $\displaystyle 2$:

$\displaystyle y=7x-10$

Remember that the slope-intercept form gives you the intercept as the final constant.  Hence, it is $\displaystyle -10$ as well!

There are two ways that you can find the y-intercept for an equation.  You could substitute $\displaystyle 0$ in for $\displaystyle x$.  This would give you:

$\displaystyle y + 84*0 = 157*0 + 250$

Simplifying, you get:

$\displaystyle y=250$

However, another way to do this is by finding the slope-intercept form of the line.  You do this by solving for $\displaystyle y$.  Indeed, this is very, very easy.  Recall that the slope intercept form is:

$\displaystyle y=mx+b$

This means that, as written, your equation obviously has $\displaystyle b=250$.  You don't even have to do all of the simplification!

In order to figure this out, you should use your slope-intercept formula.  Remember that the y-intercept is the place where $\displaystyle x$ is zero.  Therefore, the point $\displaystyle (0,10)$ gives you your y-intercept.  It is $\displaystyle 10$.  Now, to find the slope, recall the slope equation, namely:

$\displaystyle \frac{rise}{run} = \frac{y_2-y_1}{x_2-x1}$

For your points, this would be:

$\displaystyle \frac{20-10}{4-0}=\frac{10}{4}=\frac{5}{2}$

This is your slope.

Now, recall that the point-slope form of an equation is:

$\displaystyle y=mx+b$, where $\displaystyle m$ is your slope and $\displaystyle b$ is your y-intercept

Thus, your equation will be:

$\displaystyle y=\frac{5}{2}x+10$

In order to compute the slope of a line, there are several tools you can use.  For this question, try to use the slope-intercept form of a line.  Once you get the equation into this form, you basically can "read off" the slope right from the equation!  Recall that the slope-intercept form of an equation is:

$\displaystyle y=mx+b$

Now, looking at each of your options, you know that you can eliminate two immediately, as their slopes obviously are not $\displaystyle 4$:

$\displaystyle y=10x+4$

$\displaystyle y=2x+2$

The next is almost as easy:

$\displaystyle 4y-x=20$

When you solve for $\displaystyle y$, your coefficient value for $\displaystyle m$ is definitely not equal to $\displaystyle 4$:

$\displaystyle y=\frac{1}{4}x+5$

Next, $\displaystyle 3y+10x=100-50x$ is not correct either.  When you start to solve, you should notice that $\displaystyle x$ will always have a negative coefficient.  This means that it certainly will not become $\displaystyle 4$ when you finish out the simplification.

Thus, the correct answer is:

$\displaystyle 2y+2x=10x+75$

Really, all you have to pay attention to is the $\displaystyle x$ term.  First, you will subtract $\displaystyle 2x$ from both sides:

$\displaystyle 2y = 8x+75$

Then, just divide by $\displaystyle 2$, and you will have $\displaystyle 4x$! 

In order to rewrite the equation in slope-intercept form, we will need to multiply the reciprocal of the coefficient in front of y. 

$\displaystyle \frac{2y}{3} \cdot \frac{3}{2}=\frac{3}{2} (\frac{1}{2}x+3)$

Simplify both sides.

The answer is:$\displaystyle y= \frac{3}{4}x+ \frac{9}{2}$

The slope-intercept form is:  $\displaystyle y=mx+b$

Subtract $\displaystyle 3x$ on both sides.

$\displaystyle 3x-6y -3x= 10-3x$

$\displaystyle -6y = -3x+10$

Divide by negative six on both sides.

$\displaystyle \frac{-6y }{-6}= \frac{-3x+10}{-6}$

Simplify both sides.

The answer is:  $\displaystyle y= \frac{1}{2}x -\frac{5}{3}$

Slope intercept form is $\displaystyle y=mx+b$.

Add $\displaystyle 2x$ on both sides.

$\displaystyle -2x+\frac{y}{3} +2x= 2+2x$

$\displaystyle \frac{y}{3} = 2x+2$

Multiply by three on both sides.

$\displaystyle \frac{y}{3} \cdot 3 =3 (2x+2)$

The answer is:  $\displaystyle y=6x+6$