To determine how many solutions the system of linear equations has, we need to solve the system step by step. The system of equations is:
[tex]\[
\begin{array}{l}
y=\frac{1}{4} x+5 \quad \text{(1)} \\
x-4 y=4 \quad \text{(2)}
\end{array}
\][/tex]
Step 1: Substitute equation (1) into equation (2).
From equation (1), we have:
[tex]\[ y = \frac{1}{4}x + 5 \][/tex]
Substitute [tex]\( y \)[/tex] into equation (2):
[tex]\[ x - 4 \left(\frac{1}{4} x + 5\right) = 4 \][/tex]
Step 2: Simplify the equation.
Distribute [tex]\( -4 \)[/tex]:
[tex]\[ x - \left(x + 20 \right) = 4 \][/tex]
Combine like terms:
[tex]\[ x - x - 20 = 4 \][/tex]
Simplify:
[tex]\[ -20 = 4 \][/tex]
This statement [tex]\( -20 = 4 \)[/tex] is a contradiction, indicating that the system of equations is inconsistent.
Conclusion:
Since we reached a contradiction, the system of equations has no solution.
Therefore, the correct answer is:
[tex]\[ \text{No solution} \][/tex]
