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 examine what the system tells us and compare it to the provided options:

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

Step 1: Analyze the equations.

These two equations are equal to [tex]\( y \)[/tex], so we can set them equal to each other:

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

This is Option 2 from the list.

Step 2: Examine each option to see which can be derived from the system:

- Option 1: [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
This option is not derived directly from setting the two equations equal to each other, but it comes from setting the first equation [tex]\( 3x^3 - 7x^2 + 5 \)[/tex] equal to zero.

- Option 3: [tex]\( 7x^4 + 2x = 0 \)[/tex]
This option comes from setting the second equation [tex]\( 7x^4 + 2x \)[/tex] equal to zero.

- Option 4: [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
This equation combines terms from both equations by rearranging the expression [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex].

Conclusion:

The equation that is directly derived by setting the two expressions for [tex]\( y \)[/tex] equal to each other is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]

Thus, the equation that can be solved using this system is Option 2.