How do you write the equation $5x+1=4y+7$ in slope intercept form? What is the slope and y-intercept?

Hint: We start solving the problem by recalling the fact that the slope intercept form of a straight is defined as $y=mx+c$ to proceed through the problem. We then make the necessary calculations involving multiplication, addition and subtraction operations to get the required answer for the given problem. We then make use of the fact that the slope and y-intercept of the line of the form $y=mx+c$ are m and c to get the required answer.

Complete step by step answer:
According to the problem, we are asked to write the equation $5x+1=4y+7$ in its standard form.
We have given the equation as $5x+1=4y+7$.

We know that the slope intercept form of a straight is defined as $y=mx+c$. So, let us convert the given equation of line into this form.
We have $5x+1=4y+7$.
$\Rightarrow 4y=5x+1-7$.
$\Rightarrow 4y=5x-6$.
\[\Rightarrow y=\dfrac{5x-6}{4}\].
\[\Rightarrow y=\dfrac{5x}{4}-\dfrac{6}{4}\].
$\Rightarrow y=\dfrac{5}{4}x+\left( \dfrac{-3}{2} \right)$ ---(1).
So, we have found the slope intercept form of the given equation $5x+1=4y+7$ as $y=\dfrac{5}{4}x+\left( \dfrac{-3}{2} \right)$.
From the slope intercept form of the line $y=mx+c$, we know that the slope is m and y-intercept of line is c. Let us use this for obtained equation (1).
So, we get slope as $\dfrac{5}{4}$ and y-intercept as $\dfrac{-3}{2}$.

$\therefore $ The slope intercept form of the given equation $5x+1=4y+7$ as $y=\dfrac{5}{4}x+\left( \dfrac{-3}{2} \right)$ and the slope and y-intercept are $\dfrac{5}{4}$ and $\dfrac{-3}{2}$.

Note: Whenever we get this type of problem, we first recall the definition of the required equation to proceed through the solution. We can also convert the given equation of line to the standard form of the line which is $ax+by+c=0$. We should not make calculation mistakes while solving this type of problem. We should not confuse between slope and y-intercept while solving this problem. Similarly, we can expect problems to convert the given equation of line $y=2x+5$ to the standard form.