College

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 using the given system of equations, we need to analyze and compare the equations:

The system of equations given is:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]

To find a common equation from the given choices, let's follow these steps:

1. Set the two expressions for [tex]\( y \)[/tex] equal to each other:

Since both equations define [tex]\( y \)[/tex], we can set them equal to each other for comparison:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]

This equation directly matches one of the options provided:
- [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]

2. Express the equation in a standard polynomial form:

- Rearrange the equation by moving all terms to one side:
[tex]\[
0 = 7x^4 + 3x^3 - 7x^2 + 2x + 5
\][/tex]

This re-arranged equation also matches one of the provided options:
- [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]

Therefore, both of these equations can be derived from the given system of equations:
- [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
- [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]

These are the equations that can be solved using the given system.