To determine which of the given systems of equations has the solution [tex]\((1, 4)\)[/tex], we will substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex] into each system and check which one solves both equations.
System 1:
[tex]\[
\begin{array}{l}
y = -3x - 1 \\
y = -x + 5
\end{array}
\][/tex]
Substituting [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:
1. In the first equation: [tex]\(4 = -3(1) - 1 \rightarrow 4 = -3 - 1 \rightarrow 4 = -4 \quad(\text{False})\)[/tex]
2. In the second equation: [tex]\(4 = -1(1) + 5 \rightarrow 4 = -1 + 5 \rightarrow 4 = 4 \quad(\text{True})\)[/tex]
Since the first equation does not hold true, this system does not have the solution [tex]\((1, 4)\)[/tex].
System 2:
[tex]\[
\begin{array}{l}
y = 3x + 1 \\
y = -x + 5
\end{array}
\][/tex]
Substituting [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:
1. In the first equation: [tex]\(4 = 3(1) + 1 \rightarrow 4 = 3 + 1 \rightarrow 4 = 4 \quad(\text{True})\)[/tex]
2. In the second equation: [tex]\(4 = -1(1) + 5 \rightarrow 4 = -1 + 5 \rightarrow 4 = 4 \quad(\text{True})\)[/tex]
Both equations hold true, so this system has the solution [tex]\((1, 4)\)[/tex].
System 3:
[tex]\[
\begin{array}{l}
y = 3x + 1 \\
y = x - 5
\end{array}
\][/tex]
Substituting [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:
1. In the first equation: [tex]\(4 = 3(1) + 1 \rightarrow 4 = 3 + 1 \rightarrow 4 = 4 \quad(\text{True})\)[/tex]
2. In the second equation: [tex]\(4 = 1 - 5 \rightarrow 4 = -4 \quad(\text{False})\)[/tex]
Since the second equation does not hold true, this system does not have the solution [tex]\((1, 4)\)[/tex].
System 4:
[tex]\[
\begin{array}{l}
y = 3x + 1 \\
y = -x - 5
\end{array}
\][/tex]
Substituting [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:
1. In the first equation: [tex]\(4 = 3(1) + 1 \rightarrow 4 = 3 + 1 \rightarrow 4 = 4 \quad(\text{True})\)[/tex]
2. In the second equation: [tex]\(4 = -1 - 5 \rightarrow 4 = -6 \quad(\text{False})\)[/tex]
Since the second equation does not hold true, this system does not have the solution [tex]\((1, 4)\)[/tex].
Thus, the only system of equations that has the solution [tex]\((1, 4)\)[/tex] is the second system:
[tex]\[
\begin{array}{l}
y = 3x + 1 \\
y = -x + 5
\end{array}
\][/tex]
