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:  \(\displaystyle y=6x+6\)