High School

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 the two equations in the system:

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

We set these two expressions for [tex]\( y \)[/tex] equal to each other to find any common solutions, which means we're equating the right sides of both equations:

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

This is the equation we're trying to solve.

Now, let's see all the options given in the problem:

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

From our analysis, the equation that accurately represents setting the two [tex]\( y \)[/tex]-expressions equal to each other is:

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

This matches the second option in the list. This equation represents solving the system as a single equation by finding the value of [tex]\( x \)[/tex] where both expressions for [tex]\( y \)[/tex] are equal.