Answer :
We start with the system of equations
[tex]$$
\begin{cases}
y = 3x^3 - 7x^2 + 5, \\
y = 7x^4 + 2x.
\end{cases}
$$[/tex]
Since both equations equal [tex]$y$[/tex], we can set the right-hand sides equal to each other:
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
Next, we move all terms to one side of the equation to obtain an equation solely in [tex]$x$[/tex]:
[tex]$$
3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0.
$$[/tex]
Rearranging the terms in descending order, we get:
[tex]$$
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0.
$$[/tex]
When comparing with the multiple-choice options, we see that option 2 corresponds to the equation
[tex]$$
3x^3 - 7x^2 + 5 - 7x^4 + 2x = 0.
$$[/tex]
Noting that this expression is just a rearrangement of the terms (the order of addition does not affect the equality), option 2 represents the equation we obtain from eliminating [tex]$y$[/tex].
Thus, the equation that can be solved using the given system is the one in 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], we can set the right-hand sides equal to each other:
[tex]$$
3x^3 - 7x^2 + 5 = 7x^4 + 2x.
$$[/tex]
Next, we move all terms to one side of the equation to obtain an equation solely in [tex]$x$[/tex]:
[tex]$$
3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0.
$$[/tex]
Rearranging the terms in descending order, we get:
[tex]$$
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0.
$$[/tex]
When comparing with the multiple-choice options, we see that option 2 corresponds to the equation
[tex]$$
3x^3 - 7x^2 + 5 - 7x^4 + 2x = 0.
$$[/tex]
Noting that this expression is just a rearrangement of the terms (the order of addition does not affect the equality), option 2 represents the equation we obtain from eliminating [tex]$y$[/tex].
Thus, the equation that can be solved using the given system is the one in option 2.
