High School

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

[tex]
\[
\left\{
\begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array}
\right.
\]
[/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 solve the problem of finding which equation can be solved by using the given system of equations, let's look at the system:

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

We're asked to determine which equation from the provided options can be derived from this system.

### Explanation

In order to find which equation is derived from the system, we look for the intersection points of the two equations. This means we set the right-hand sides of the given equations equal to each other. This is because both expressions define [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], and if they express the same [tex]\( y \)[/tex], they must be equal at particular values of [tex]\( x \)[/tex].

1. Set the equations equal to each other:

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

This gives us the equation option:

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

This equation compares both expressions for [tex]\( y \)[/tex] from the given system, checking where they have the same values, which is essentially finding the intersection points of the curves represented by the system. Therefore, the equation [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex] is the correct one that can be solved using the given system of equations.