Solve the system of equations:
[tex]\[
\begin{array}{l}
y = x + 4 \\
y = -2x - 5
\end{array}
\][/tex]

Options:
A. [tex]$(-4, 0)$[/tex]
B. [tex]$(-3, 1)$[/tex]
C. [tex]$(0, -5)$[/tex]
D. [tex]$(-2, -1)$[/tex]



Answer :

To determine which of the given points satisfy both equations [tex]\( y = x + 4 \)[/tex] and [tex]\( y = -2x - 5 \)[/tex], we will substitute the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values from each point into both equations and check for consistency.

### Checking Point [tex]\((-4, 0)\)[/tex]:

1. Substituting [tex]\( x = -4 \)[/tex] into [tex]\( y = x + 4 \)[/tex]:
[tex]\[ y = -4 + 4 = 0 \][/tex]
This is consistent with the [tex]\( y \)[/tex] value of the point.

2. Substituting [tex]\( x = -4 \)[/tex] into [tex]\( y = -2x - 5 \)[/tex]:
[tex]\[ y = -2(-4) - 5 = 8 - 5 = 3 \][/tex]
This does not match the [tex]\( y \)[/tex] value of the point.

Hence, [tex]\((-4, 0)\)[/tex] does not satisfy both equations.

### Checking Point [tex]\((-3, 1)\)[/tex]:

1. Substituting [tex]\( x = -3 \)[/tex] into [tex]\( y = x + 4 \)[/tex]:
[tex]\[ y = -3 + 4 = 1 \][/tex]
This is consistent with the [tex]\( y \)[/tex] value of the point.

2. Substituting [tex]\( x = -3 \)[/tex] into [tex]\( y = -2x - 5 \)[/tex]:
[tex]\[ y = -2(-3) - 5 = 6 - 5 = 1 \][/tex]
This is also consistent with the [tex]\( y \)[/tex] value of the point.

Hence, [tex]\((-3, 1)\)[/tex] satisfies both equations.

### Checking Point [tex]\((0, -5)\)[/tex]:

1. Substituting [tex]\( x = 0 \)[/tex] into [tex]\( y = x + 4 \)[/tex]:
[tex]\[ y = 0 + 4 = 4 \][/tex]
This does not match the [tex]\( y \)[/tex] value of the point.

2. Substituting [tex]\( x = 0 \)[/tex] into [tex]\( y = -2x - 5 \)[/tex]:
[tex]\[ y = -2(0) - 5 = -5 \][/tex]
This matches the [tex]\( y \)[/tex] value of the point.

Hence, [tex]\((0, -5)\)[/tex] does not satisfy both equations.

### Checking Point [tex]\((-2, -1)\)[/tex]:

1. Substituting [tex]\( x = -2 \)[/tex] into [tex]\( y = x + 4 \)[/tex]:
[tex]\[ y = -2 + 4 = 2 \][/tex]
This does not match the [tex]\( y \)[/tex] value of the point.

2. Substituting [tex]\( x = -2 \)[/tex] into [tex]\( y = -2x - 5 \)[/tex]:
[tex]\[ y = -2(-2) - 5 = 4 - 5 = -1 \][/tex]
This matches the [tex]\( y \)[/tex] value of the point.

Hence, [tex]\((-2, -1)\)[/tex] does not satisfy both equations.

### Conclusion

Among the points provided, only [tex]\((-3, 1)\)[/tex] satisfies both equations [tex]\( y = x + 4 \)[/tex] and [tex]\( y = -2x - 5 \)[/tex]. Thus, the point [tex]\((-3, 1)\)[/tex] is the solution.