Answer :
To solve the given system of equations, you need to find the common solutions for both expressions, which means you would set the two equations equal to each other.
Here's a step-by-step guide to understanding and solving the system of equations:
1. Identify the System of Equations:
[tex]\[
\begin{align*}
y &= 3x^3 - 7x^2 + 5 \\
y &= 7x^4 + 2x
\end{align*}
\][/tex]
2. Set the Equations Equal:
In order for both equations to be true simultaneously (since they both equal [tex]\(y\)[/tex]), set them equal to each other:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]
3. Rearrange the Equation:
Subtract the right side from the left side to consolidate terms:
[tex]\[
3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0
\][/tex]
4. Simplify the Equation:
Combine like terms to get a single polynomial equation:
[tex]\[
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0
\][/tex]
5. Conclusion:
You have simplified the system of equations to a single polynomial equation. The correct answer is:
[tex]\[
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0
\][/tex]
This equation can be solved to find the values of [tex]\(x\)[/tex] that satisfy the original system. Among the given options, this equation matches the one presented above:
[tex]\[
7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0
\][/tex]
Note: Switching the signs in rearranging or simplifying doesn't fundamentally change the problem since you can multiply through by [tex]\(-1\)[/tex] if needed, keeping the equation equivalent. Therefore, check both formats when given choices.
Here's a step-by-step guide to understanding and solving the system of equations:
1. Identify the System of Equations:
[tex]\[
\begin{align*}
y &= 3x^3 - 7x^2 + 5 \\
y &= 7x^4 + 2x
\end{align*}
\][/tex]
2. Set the Equations Equal:
In order for both equations to be true simultaneously (since they both equal [tex]\(y\)[/tex]), set them equal to each other:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]
3. Rearrange the Equation:
Subtract the right side from the left side to consolidate terms:
[tex]\[
3x^3 - 7x^2 + 5 - 7x^4 - 2x = 0
\][/tex]
4. Simplify the Equation:
Combine like terms to get a single polynomial equation:
[tex]\[
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0
\][/tex]
5. Conclusion:
You have simplified the system of equations to a single polynomial equation. The correct answer is:
[tex]\[
-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0
\][/tex]
This equation can be solved to find the values of [tex]\(x\)[/tex] that satisfy the original system. Among the given options, this equation matches the one presented above:
[tex]\[
7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0
\][/tex]
Note: Switching the signs in rearranging or simplifying doesn't fundamentally change the problem since you can multiply through by [tex]\(-1\)[/tex] if needed, keeping the equation equivalent. Therefore, check both formats when given choices.
