High School

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

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

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 find which equation can be solved using the given system of equations:

1. Understand the System of Equations:
We have a system of two equations:
- [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
- [tex]\( y = 7x^4 + 2x \)[/tex]

2. Identify the Method to Combine Equations:
Since both equations equal [tex]\( y \)[/tex], we can set them equal to each other to find a relationship involving only [tex]\( x \)[/tex].

3. Set the Equations Equal:
By substituting [tex]\( y \)[/tex] from the first equation into the second, or vice-versa, we equate the two expressions:
- [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]

4. Recognize the Resulting Equation:
The equation [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex] can be solved for [tex]\( x \)[/tex], as it originated by equating the two expressions given by the system.

Based on these steps, the equation that can be solved using the given system of equations is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]