Find all solutions of the given system of equations and check your answer gr answer in terms of \( x \), where \( y=y(x) \).) \( \left.\qquad \begin{array}{l}4 x+5 y=1 \\ -x-\frac{5}{4} y=-\frac{1}{4} \\ (x, y)=(\text { NOSOLUTION }\end{array}\right) \)

To find the solutions of the given system of equations, we can analyze them step by step. We have the following equations: 1. \( 4x + 5y = 1 \) (Equation 1) 2. \( -x - \frac{5}{4}y = -\frac{1}{4} \) (Equation 2) First, let's rewrite Equation 2 for clarity. Multiplying through by -4 to eliminate the fraction gives us: \[ 4x + 5y = 1 \] This is actually the same as Equation 1. Since both equations represent the same line, they are dependent, meaning that there are infinitely many solutions along this line. To express \( y \) in terms of \( x \), we can rearrange Equation 1: \[ 5y = 1 - 4x \] Dividing by 5 gives: \[ y = \frac{1}{5} - \frac{4}{5}x \] Thus, the solutions in terms of \( x \) are: \[ (x,y) = \left( x, \frac{1}{5} - \frac{4}{5}x \right) \] This means there is an infinite number of solutions along the line defined by the rewritten equation. Since both equations are identical, we conclude that there are no contradictions or unique solutions. So, we summarize that the system has infinitely many solutions, with the relationship between \( x \) and \( y \) given by: \[ y = \frac{1}{5} - \frac{4}{5}x \]