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, let's take a closer look at what we have:

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

To find a relation between these equations, let's set them equal to each other since they both equal [tex]\( y \)[/tex]:

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

Now, let's analyze the options given in the question:

1. [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]: This equation is derived from setting the first equation equal to zero, which doesn’t directly solve the system as a whole.

2. [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]: This equation is exactly the one we obtained by equating the two expressions for [tex]\( y \)[/tex]. So this is the equation that directly represents solving the system.

3. [tex]\( 7x^4 + 2x = 0 \)[/tex]: This equation comes from setting just the second equation equal to zero, which is not the goal for solving the system.

4. [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]: This equation appears to rearrange and combine terms in a way that doesn’t match simply equating and solving the given system.

Thus, the equation that can be solved using this system is:

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

The correct answer is choice 2.