College

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 determine which equation can be solved using the given system of equations, let's look closely at the system:

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

Since both equations equal [tex]\( y \)[/tex], we can set these two expressions equal to each other to find a relationship between them. This leads to the equation:

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

This equation matches one of the options provided:

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

The equation [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex] corresponds to the second option. This is the equation derived directly from equating the two given functions, meaning it can indeed be solved using the system of equations.