Answer :
To determine which equation can be solved by using the given system of equations, let's carefully analyze them:
```
\left\{\begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array}\right.
```
To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we should set the two expressions for [tex]\( y \)[/tex] equal to each other because they both represent [tex]\( y \)[/tex].
So, we equate:
[tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
This gives us one of the possible equations:
Equation:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Now, let's check the rest of the options one by one:
1. Option:
[tex]\[ 3x^3 - 7x^2 + 5 = 0 \][/tex]
This equation comes directly from setting the first equation [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex] equal to 0. However, it doesn't take into account the second equation in the system.
2. Option:
[tex]\[ 7x^4 + 2x = 0 \][/tex]
This is derived from setting the second equation [tex]\( y = 7x^4 + 2x \)[/tex] equal to 0, which also does not involve the first equation.
3. Option:
[tex]\[ 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \][/tex]
This is a more complex polynomial that includes terms from both original equations but doesn't align correctly with the system we were given.
The correct equation, which comes directly from setting the two expressions for [tex]\( y \)[/tex] equal to each other from the system, is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Hence, the equation that can be solved using the given system of equations is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x. \][/tex]
The correct option is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x. \][/tex]
```
\left\{\begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array}\right.
```
To solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we should set the two expressions for [tex]\( y \)[/tex] equal to each other because they both represent [tex]\( y \)[/tex].
So, we equate:
[tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
This gives us one of the possible equations:
Equation:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Now, let's check the rest of the options one by one:
1. Option:
[tex]\[ 3x^3 - 7x^2 + 5 = 0 \][/tex]
This equation comes directly from setting the first equation [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex] equal to 0. However, it doesn't take into account the second equation in the system.
2. Option:
[tex]\[ 7x^4 + 2x = 0 \][/tex]
This is derived from setting the second equation [tex]\( y = 7x^4 + 2x \)[/tex] equal to 0, which also does not involve the first equation.
3. Option:
[tex]\[ 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \][/tex]
This is a more complex polynomial that includes terms from both original equations but doesn't align correctly with the system we were given.
The correct equation, which comes directly from setting the two expressions for [tex]\( y \)[/tex] equal to each other from the system, is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Hence, the equation that can be solved using the given system of equations is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x. \][/tex]
The correct option is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x. \][/tex]
