Answer :
We start with the system
[tex]$$
\begin{cases}
y = 3x^3 - 7x^2 + 5, \\
y = 7x^4 + 2x.
\end{cases}
$$[/tex]
Since both equations equal [tex]$y$[/tex], the intersection points occur when
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
This is the equation from which we need to solve for [tex]$x$[/tex]. Therefore, the correct choice is the equation
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
This corresponds to option 2.
For completeness, if we were to rearrange the equation, we subtract [tex]$7x^4 + 2x$[/tex] from both sides to obtain
[tex]$$
3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0.
$$[/tex]
Reordering the terms gives
[tex]$$
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0.
$$[/tex]
Multiplying by [tex]$-1$[/tex] for a positive leading coefficient results in
[tex]$$
7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0.
$$[/tex]
This is an equivalent form, but the equation that was directly set up from the system is
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
Thus, the answer is option 2.
[tex]$$
\begin{cases}
y = 3x^3 - 7x^2 + 5, \\
y = 7x^4 + 2x.
\end{cases}
$$[/tex]
Since both equations equal [tex]$y$[/tex], the intersection points occur when
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
This is the equation from which we need to solve for [tex]$x$[/tex]. Therefore, the correct choice is the equation
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
This corresponds to option 2.
For completeness, if we were to rearrange the equation, we subtract [tex]$7x^4 + 2x$[/tex] from both sides to obtain
[tex]$$
3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0.
$$[/tex]
Reordering the terms gives
[tex]$$
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0.
$$[/tex]
Multiplying by [tex]$-1$[/tex] for a positive leading coefficient results in
[tex]$$
7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0.
$$[/tex]
This is an equivalent form, but the equation that was directly set up from the system is
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
Thus, the answer is option 2.
