To solve the system of equations and find the value of [tex]\( n = (x + y)^3 \)[/tex], we need to follow these steps:
1. Define the equations:
[tex]\[
x^2 + y^3 = 14
\][/tex]
[tex]\[
xy = 5
\][/tex]
2. Solve the system of equations:
To find the possible values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we use algebraic techniques to solve the system. However, it turns out that this system does not produce any real number solutions [tex]\((x, y)\)[/tex] that satisfy both equations simultaneously.
3. Determine [tex]\( (x + y)^3 \)[/tex]:
Since there are no [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values that satisfy both given equations, the calculation of [tex]\( (x + y)^3 \)[/tex] can not proceed with any valid pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
4. Conclusion:
Because no valid solutions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] exist, we conclude that there are no valid values for [tex]\( (x + y)^3 \)[/tex].
In conclusion, the value of [tex]\( n = (x + y)^3 \)[/tex] given the system of equations is:
[tex]\[
\boxed{[]}
\][/tex]
There are no values of [tex]\( n \)[/tex] that satisfy the system of equations.
