To determine the number of solutions for the system of equations:
[tex]\[
\begin{array}{l}
y = -\frac{1}{3} x + 7 \\
y = -2 x^3 + 5 x^2 + x - 2
\end{array}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] where the two equations intersect, i.e., where they are equal to each other. This translates to solving the equation:
[tex]\[
-\frac{1}{3} x + 7 = -2 x^3 + 5 x^2 + x - 2
\][/tex]
Bringing all terms to one side to set the equation to zero, we obtain:
[tex]\[
0 = -2x^3 + 5x^2 + x - 2 + \frac{1}{3}x - 7
\][/tex]
Simplifying the terms results in:
[tex]\[
0 = -2x^3 + 5x^2 + \left( x + \frac{1}{3}x \right) - 2 - 7
\][/tex]
Combining like terms:
[tex]\[
0 = -2x^3 + 5x^2 + \frac{4}{3}x - 9
\][/tex]
Since this is a cubic polynomial equation, it's known that a cubic polynomial can have up to three roots (including real and complex roots). Examining the solution carefully, we find that there are three roots for this polynomial, two of which are complex and one real.
From the result provided, we have three solutions for [tex]\( x \)[/tex]:
[tex]\[
x_1 = -1.19709411413168 \quad \text{(real)}
\][/tex]
[tex]\[
x_2 = 1.84854705706584 - 0.584787751722331i \quad \text{(complex)}
\][/tex]
[tex]\[
x_3 = 1.84854705706584 + 0.584787751722331i \quad \text{(complex)}
\][/tex]
Therefore, the system of equations has three solutions. The correct answer is:
D. 3 solutions
