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Which of the following systems of equations has the solution [tex]$(1,4)$[/tex]?

A.
[tex]
\begin{array}{l}
y = -3x - 1 \\
y = -x + 5
\end{array}
[/tex]

B.
[tex]
\begin{array}{l}
y = 3x + 1 \\
y = -x + 5
\end{array}
[/tex]

C.
[tex]
\begin{array}{l}
y = 3x + 1 \\
y = x - 5
\end{array}
[/tex]

D.
[tex]
\begin{array}{l}
y = 3x + 1 \\
y = -x - 5
\end{array}
[/tex]

Sagot :

GinnyAnsweridn
To determine which system of equations has the solution [tex]\((1, 4)\)[/tex], we can substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex] into each equation and verify if both equations are satisfied.

### System 1:
[tex]\[ \begin{cases} y = -3x - 1 \\ y = -x + 5 \end{cases} \][/tex]

Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:

First equation:
[tex]\[ 4 = -3(1) - 1 \][/tex]
[tex]\[ 4 = -3 - 1 \][/tex]
[tex]\[ 4 = -4 \][/tex] (which is false)

Since the first equation is not satisfied, this system does not have the solution [tex]\((1, 4)\)[/tex].

### System 2:
[tex]\[ \begin{cases} y = 3x + 1 \\ y = -x + 5 \end{cases} \][/tex]

Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:

First equation:
[tex]\[ 4 = 3(1) + 1 \][/tex]
[tex]\[ 4 = 3 + 1 \][/tex]
[tex]\[ 4 = 4 \][/tex] (which is true)

Second equation:
[tex]\[ 4 = -(1) + 5 \][/tex]
[tex]\[ 4 = -1 + 5 \][/tex]
[tex]\[ 4 = 4 \][/tex] (which is also true)

Both equations are satisfied, so this system does have the solution [tex]\((1, 4)\)[/tex].

### System 3:
[tex]\[ \begin{cases} y = 3x + 1 \\ y = x - 5 \end{cases} \][/tex]

Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:

First equation:
[tex]\[ 4 = 3(1) + 1 \][/tex]
[tex]\[ 4 = 3 + 1 \][/tex]
[tex]\[ 4 = 4 \][/tex] (which is true)

Second equation:
[tex]\[ 4 = 1 - 5 \][/tex]
[tex]\[ 4 = -4 \][/tex] (which is false)

Since the second equation is not satisfied, this system does not have the solution [tex]\((1, 4)\)[/tex].

### System 4:
[tex]\[ \begin{cases} y = 3x + 1 \\ y = -x - 5 \end{cases} \][/tex]

Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 4\)[/tex]:

First equation:
[tex]\[ 4 = 3(1) + 1 \][/tex]
[tex]\[ 4 = 3 + 1 \][/tex]
[tex]\[ 4 = 4 \][/tex] (which is true)

Second equation:
[tex]\[ 4 = -(1) - 5 \][/tex]
[tex]\[ 4 = -1 - 5 \][/tex]
[tex]\[ 4 = -6 \][/tex] (which is false)

Since the second equation is not satisfied, this system does not have the solution [tex]\((1, 4)\)[/tex].

### Conclusion:
The system that has the solution [tex]\((1, 4)\)[/tex] is:
[tex]\[ \begin{cases} y = 3x + 1 \\ y = -x + 5 \end{cases} \][/tex]
So, the answer is the second system of equations.
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