High School

Which equation can be solved by using this system of equations?

[tex]
\[
\begin{cases}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{cases}
\]
[/tex]

A. [tex]\(3x^3 - 7x^2 + 5 = 0\)[/tex]

B. [tex]\(3x^3 - 7x^2 + 5 = 7x^4 + 2x\)[/tex]

C. [tex]\(7x^4 + 2x = 0\)[/tex]

D. [tex]\(7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0\)[/tex]

Answer :

To determine which equation can be solved by using the given system of equations, we have:

1. The first equation is [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex].
2. The second equation is [tex]\( y = 7x^4 + 2x \)[/tex].

Since both expressions are equal to [tex]\( y \)[/tex], we can set them equal to each other to find a suitable equation. This strategy helps us identify where both expressions for [tex]\( y \)[/tex] are equal, representing the point(s) of intersection of the two curves.

Let's set the two expressions equal:

[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]

This equation, [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex], is what we're looking for because it arises directly from equating the two expressions of [tex]\( y \)[/tex].

Therefore, the equation that can be solved using this system of equations is:

[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]