The student was correct in the attempt to get the equation into slope-intercept form, $\displaystyle y = mx + b$ by dividing by $\displaystyle 3$ on both sides.

The slope should have been: $\displaystyle m = \frac{4}{3}$

The y-intercept was correct in being: $\displaystyle b = 7$

Start by finding the slope of the line.

Recall how to find the slope:

$\displaystyle \text{Slope}=\frac{y_2-y_1}{x_2-x_1}$

Using the given points,

$\displaystyle \text{Slope}=\frac{-6-6}{2-(-1)}=\frac{-12}{3}=-4$

Now, we can write the equation for the line as the following:

$\displaystyle y=-4x+b$, where $\displaystyle b$ is the y-intercept that we still need to find.

Take one of the points and plug it into the equation for $\displaystyle x$ and $\displaystyle y$, then solve for $\displaystyle b$.

Using the point $\displaystyle (-1, 6)$,

$\displaystyle 6=-4(-1)+b$

$\displaystyle 6=4+b$

$\displaystyle b=2$

Thus, the equation of the line must be $\displaystyle y=-4x+2$

So our final answer should appear in slope-intercept form, $\displaystyle \small y=mx+b$ with $\displaystyle \small m$ representing the slope and $\displaystyle \small b$ representing the y-intercept. We know that our slope is $\displaystyle \small -\frac{2}{3}$, meaning $\displaystyle \small m=-\frac{2}{3}$. 

Now we have $\displaystyle \small y=-\frac{2}{3}x+b$ but we still need to find our y-intercept, $\displaystyle \small b$. 

To solve for the y-intercept, we'll need to use the coordinates given to us in the question to replace the $\displaystyle \small x$ and $\displaystyle \small y$. Remember that in a coordinate the $\displaystyle \small x$ is our first number and our $\displaystyle \small y$ is the second number, like so: $\displaystyle \small (x,y)$.

Since we are working with fractions here i'll show how to solve this without a calculator, but using one will make it quicker.

Replace the $\displaystyle \small x$ and y with $\displaystyle \small -2$ and $\displaystyle \small -3$ respectively and then solve as if you solving for $\displaystyle \small x$, but with $\displaystyle \small b$.

$\displaystyle \small -3=-\frac{2}{3}-2+b$

Since we are multiplying with a fraction, our $\displaystyle \small -2$ can be changed to look like $\displaystyle \small -\frac{2}{1}$, which is $\displaystyle \small -2$'s fraction form. Multiply across both the top and bottom.

$\displaystyle \small -\frac{2}{3}\cdot-\frac{2}{1}=\frac{4}{3}$

So now we have this:

$\displaystyle \small -3=\frac{4}{3}+b$

Subtract the $\displaystyle \small \frac{4}{3}$ on both sides, and since we're subtracting by a fraction we'll need our $\displaystyle \small -3$ to become a fraction too. We can't use $\displaystyle \small -\frac{3}{1}$ because for adding and subtracting our denominators must be the same, so I will multiply $\displaystyle \small -\frac{3}{1}$ with $\displaystyle \small 3$ in order to get the same denominator. 

$\displaystyle \small -\frac{3}{1}\cdot3=-\frac{9}{3}$

Now that our $\displaystyle \small -3$ has become $\displaystyle \small -\frac{9}{3}$ (it's still $\displaystyle \small -3$, despite how big the fraction looks.) we can use it with our subtraction of $\displaystyle \small \frac{4}{3}$. Subtract only the numerator though, not the denominator.

$\displaystyle \small -\frac{9}{3}-\frac{4}{3}=-\frac{13}{3}$

$\displaystyle \small -\frac{13}{3}=b$

Now that we have our y-intercept, we can take out the $\displaystyle \small -2$ and $\displaystyle \small -3$ and replace our $\displaystyle \small b$ with $\displaystyle \small -\frac{13}{3}$.

$\displaystyle \small y=-\frac{2}{3}x-\frac{13}{3}$

Our answer should be in slope-intercept form, $\displaystyle y=mx+b$ with $\displaystyle \small m$ representing our slope and $\displaystyle \small b$ representing our y-intercept. We know that our slope is$\displaystyle -3$, which means $\displaystyle m=-3$. 

This should give us $\displaystyle y=-3x+b$, but we still need to find our y-intercept; $\displaystyle \small b$.

In order to find our y-intercept, we'll need to replace our $\displaystyle \small x$ and $\displaystyle \small y$ with those of our coordinates in the question. Remember that in a coordinate the first number is our $\displaystyle \small x$ while our second number is $\displaystyle \small y$, as shown here: $\displaystyle (x,y)$.

Replace $\displaystyle \small x$ and $\displaystyle \small y$ with that of $\displaystyle -3$ and $\displaystyle \small 12$ and then solve the problem as if you were solving for $\displaystyle \small x$, but with $\displaystyle \small b$.

$\displaystyle 12=(-3)-3+b$

Both negatives when multiplied cancel to create a positive:

$\displaystyle 12=9+b$

Subtract $\displaystyle \small 9$ from both sides:

$\displaystyle 12-9=b$

$\displaystyle 3=b$

Our y-intercept is $\displaystyle \small 3$, so now we can take out the $\displaystyle \small 12$ and $\displaystyle -3$ and replace the $\displaystyle \small b$ with $\displaystyle \small +3$.

$\displaystyle y=-3x+3$

So we know we need this problem to end as a slope-intercept formula, $\displaystyle y=mx+b$ with $\displaystyle \small m$ representing our slope and $\displaystyle \small b$ representing our y-intercept.

From the question we know that our slope is $\displaystyle \small 2$, which means $\displaystyle m=2$. So we have $\displaystyle y=2x+b$ so far, now we need to find our y-intercept; $\displaystyle \small b$.

To find $\displaystyle \small b$, you need to plug in our coordinates $\displaystyle (3,6)$ into the equation. Remember that the first number of a coordinate is your $\displaystyle \small x$, and the second one is your $\displaystyle \small y$, like this $\displaystyle (x,y)$.

Take the $\displaystyle \small 3$ and $\displaystyle \small 6$ of the coordinate and substitute them for your $\displaystyle \small x$ and $\displaystyle \small y$, so you should end up with something looking like this: $\displaystyle 6=(2)3+b$

Solve the problem from there like you would to find $\displaystyle \small x$, only with $\displaystyle \small b$.

$\displaystyle 6=6+b$

$\displaystyle 6-6=b$

$\displaystyle 0=b$

Our y-intercept is $\displaystyle \small 0$, so now we can take out the $\displaystyle \small 3$ and $\displaystyle \small 6$ and substitute the $\displaystyle \small b$ for $\displaystyle \small 0$. 

$\displaystyle y=2x+0$

The slope-intercept form of the equation of a line is 

$\displaystyle y = mx+ b$ 

for some constant $\displaystyle m,b$.

To rewrite 

$\displaystyle 6x + 7y = 87$

in this form, it is necessary to solve for $\displaystyle y$, isolating it on the left-side. First, add $\displaystyle -6x$ to both sides:

$\displaystyle 6x + 7y + (-6x) = 87 + (-6x)$

$\displaystyle 7y =-6x+ 87$

Multiply both sides by $\displaystyle \frac{1}{7}$:

$\displaystyle \frac{1}{7} \cdot 7y =\frac{1}{7} \cdot (-6x+ 87)$

Distribute on the right:

$\displaystyle y =\frac{1}{7} \cdot (-6x)+ \frac{1}{7} \cdot (87)$

$\displaystyle y =-\frac{6}{7} x + \frac{87}{7}$

This is the correct choice.

Recall what the slope intercept form is:

$\displaystyle y=mx+b$

You will need to algebraically rearrange the given equation.

$\displaystyle 5x+7y=12$

$\displaystyle 7y=12-5x$

$\displaystyle y=-\frac{5}{7}x+\frac{12}{7}$ is the slope-intercept form of the equation given in standard form.

Recall our point-slope form

$\displaystyle y-y_1=m(x-x_1)$

Here $\displaystyle y_1=4$ and $\displaystyle x_1=3$ and $\displaystyle m=2$

So, plugging those in gives us

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

Lets distribute that 2

$\displaystyle y-4=2x-6$

and add 4 to both sides

$\displaystyle y-4+4=2x-6+4$

And simplify

$\displaystyle y=2x-2$