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 solve this question, we need to determine which equation can be solved using the provided system of equations:

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

2. Understanding the System:
Both equations are expressions for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. To find a common solution for this system, we need to set these two expressions for [tex]\( y \)[/tex] equal to each other since they both represent the same variable:

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

3. Which Equation Solves the System?:
The equation [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex] is the one that directly arises from setting the two expressions equal to each other. Solving this equation will yield the values of [tex]\( x \)[/tex] for which the two expressions for [tex]\( y \)[/tex] will have the same value.

Therefore, the equation we can solve based on this system is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]

This equation represents the step necessary to find solutions where both expressions for [tex]\( y \)[/tex] are equal for some values of [tex]\( x \)[/tex].