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Answer :
Sure! Let's solve this step-by-step.
We are given the system of equations:
[tex]\[
\left\{ \begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array} \right.
\][/tex]
To find an equation that can be solved using this system, we need to eliminate [tex]\( y \)[/tex] by equating the two expressions given for [tex]\( y \)[/tex].
1. We start by setting the right-hand sides of the equations equal to each other:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
2. Next, we move all the terms to one side of the equation to set it equal to zero:
[tex]\[ 3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0 \][/tex]
3. Now we combine like terms to simplify the equation:
[tex]\[ -7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0 \][/tex]
We can rearrange this for a more conventional form:
[tex]\[ 7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0 \][/tex]
So, the equation that can be solved by using the given system of equations is:
[tex]\[ 7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0 \][/tex]
This matches the fourth option provided in the question:
[tex]\[ 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \][/tex]
Therefore, this is the correct equation to solve with the given system of equations.
We are given the system of equations:
[tex]\[
\left\{ \begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array} \right.
\][/tex]
To find an equation that can be solved using this system, we need to eliminate [tex]\( y \)[/tex] by equating the two expressions given for [tex]\( y \)[/tex].
1. We start by setting the right-hand sides of the equations equal to each other:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
2. Next, we move all the terms to one side of the equation to set it equal to zero:
[tex]\[ 3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0 \][/tex]
3. Now we combine like terms to simplify the equation:
[tex]\[ -7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0 \][/tex]
We can rearrange this for a more conventional form:
[tex]\[ 7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0 \][/tex]
So, the equation that can be solved by using the given system of equations is:
[tex]\[ 7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0 \][/tex]
This matches the fourth option provided in the question:
[tex]\[ 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \][/tex]
Therefore, this is the correct equation to solve with the given system of equations.
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