Let's go through each equation and determine the number of solutions.
1. Equation: [tex]\( y = -6x - 9 \)[/tex]
This is a linear equation. A linear equation of the form [tex]\( y = mx + b \)[/tex] has exactly one solution.
- Number of Solutions: 1
2. Equation: [tex]\( y = 2x^5 - x^4 + 2x^3 - 6x^2 + 2x + 4 \)[/tex]
This is a polynomial equation of degree 5. A polynomial equation of degree [tex]\( n \)[/tex] can have up to [tex]\( n \)[/tex] solutions, depending on real and complex roots. In this case, this equation has exactly 5 solutions.
- Number of Solutions: 5
3. Equation: [tex]\( y = -4x^3 - 10x^2 + 7x + 5 \)[/tex]
This is a polynomial equation of degree 3. A polynomial equation of degree [tex]\( n \)[/tex] can have up to [tex]\( n \)[/tex] solutions. In this case, this equation has exactly 3 solutions.
- Number of Solutions: 3
Summarizing:
[tex]\[
\begin{tabular}{|l|c|}
\hline \multicolumn{1}{|c|}{ Equation } & \begin{tabular}{c}
Number of \\
Solutions
\end{tabular} \\
\hline$y=-6 x-9$ & 1 \\
\hline$y=2 x^5-x^4+2 x^3-6 x^2+2 x+4$ & 5 \\
\hline$y=-4 x^3-10 x^2+7 x+5$ & 3 \\
\hline
\end{tabular}
\][/tex]
