To determine the solution of the given system of equations:
[tex]\[
\begin{array}{c}
2x - 5y = 8 \\
5y = -2x + 8
\end{array}
\][/tex]
First, let's rewrite the second equation in a more standard form:
[tex]\[
5y = -2x + 8 \implies -2x + 5y = 8
\][/tex]
So the system is:
[tex]\[
\begin{array}{c}
2x - 5y = 8 \\
-2x + 5y = 8
\end{array}
\][/tex]
Next, we observe the coefficients of the variables and the constants:
[tex]\[
\begin{array}{ccc}
2x - 5y &=& 8 \\
-2x + 5y &=& 8
\end{array}
\][/tex]
To determine if this system has solutions, we should consider the following steps:
1. Calculate the determinant of the coefficient matrix:
[tex]\[
\text{Determinant} = (2)(5) - (-2)(-5)
\][/tex]
[tex]\[
\text{Determinant} = 10 - 10 = 0
\][/tex]
Since the determinant is zero, the coefficient matrix is singular, which means the system of equations could either have no solutions or infinitely many solutions.
2. Check if the equations are proportional:
We need to verify if the ratios of the coefficients match the ratio of the constants:
[tex]\[
\frac{2}{-2} = \frac{-5}{5} = \frac{8}{8}
\][/tex]
[tex]\[
-1 = -1 = 1
\][/tex]
These ratios do not match, implying that the equations are not proportional and thus do not represent the same line. Therefore, the system of equations is inconsistent.
Conclusion:
The system of equations has no solutions.
So, the best way to describe the solution of this system of equations is:
There are no solutions of the system.
