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.
