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Sagot :
To find the solutions to the system of equations:
1. [tex]\( y = (x + 4)^2 - 1 \)[/tex]
2. [tex]\( y = -3x - 13 \)[/tex]
we need to find the points [tex]\((x, y)\)[/tex] that satisfy both equations.
### Step-by-Step Solution:
1. Set the equations equal to each other:
Since both expressions equal [tex]\( y \)[/tex], we can set them equal to each other:
[tex]\[ (x + 4)^2 - 1 = -3x - 13 \][/tex]
2. Simplify the equation:
Expand and simplify the equation:
[tex]\[ (x + 4)^2 - 1 = -3x - 13 \][/tex]
[tex]\[ x^2 + 8x + 16 - 1 = -3x - 13 \][/tex]
[tex]\[ x^2 + 8x + 15 = -3x - 13 \][/tex]
Move all terms to one side:
[tex]\[ x^2 + 11x + 28 = 0 \][/tex]
3. Solve the quadratic equation:
We solve the quadratic equation [tex]\( x^2 + 11x + 28 = 0 \)[/tex].
4. Roots of the quadratic equation:
The roots of the quadratic equation [tex]\(x^2 + 11x + 28 = 0\)[/tex] are [tex]\(x = -4\)[/tex] and [tex]\(x = -7\)[/tex].
5. Calculate the corresponding [tex]\( y \)[/tex] values:
For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = (x + 4)^2 - 1 \][/tex]
[tex]\[ y = (-4 + 4)^2 - 1 \][/tex]
[tex]\[ y = 0^2 - 1 \][/tex]
[tex]\[ y = -1 \][/tex]
For [tex]\( x = -7 \)[/tex]:
[tex]\[ y = (x + 4)^2 - 1 \][/tex]
[tex]\[ y = (-7 + 4)^2 - 1 \][/tex]
[tex]\[ y = (-3)^2 - 1 \][/tex]
[tex]\[ y = 9 - 1 \][/tex]
[tex]\[ y = 8 \][/tex]
Therefore, the points that satisfy both equations are [tex]\((-4, -1)\)[/tex] and [tex]\((-7, 8)\)[/tex].
6. Check the solutions against the given options:
- [tex]\((-7, -4)\)[/tex] and [tex]\((8, -1)\)[/tex] - No, these do not match our solutions.
- [tex]\((-7, 8)\)[/tex] and [tex]\((-4, -1)\)[/tex] - Yes, these match our solutions.
- [tex]\((-4, -7)\)[/tex] and [tex]\((-1, 8)\)[/tex] - No, these do not match our solutions.
- [tex]\((8, -7)\)[/tex] and [tex]\((-1, -4)\)[/tex] - No, these do not match our solutions.
Thus, the correct pair of points that solve the system of equations is:
[tex]\[ \boxed{(-7, 8) \text{ and } (-4, -1)} \][/tex]
1. [tex]\( y = (x + 4)^2 - 1 \)[/tex]
2. [tex]\( y = -3x - 13 \)[/tex]
we need to find the points [tex]\((x, y)\)[/tex] that satisfy both equations.
### Step-by-Step Solution:
1. Set the equations equal to each other:
Since both expressions equal [tex]\( y \)[/tex], we can set them equal to each other:
[tex]\[ (x + 4)^2 - 1 = -3x - 13 \][/tex]
2. Simplify the equation:
Expand and simplify the equation:
[tex]\[ (x + 4)^2 - 1 = -3x - 13 \][/tex]
[tex]\[ x^2 + 8x + 16 - 1 = -3x - 13 \][/tex]
[tex]\[ x^2 + 8x + 15 = -3x - 13 \][/tex]
Move all terms to one side:
[tex]\[ x^2 + 11x + 28 = 0 \][/tex]
3. Solve the quadratic equation:
We solve the quadratic equation [tex]\( x^2 + 11x + 28 = 0 \)[/tex].
4. Roots of the quadratic equation:
The roots of the quadratic equation [tex]\(x^2 + 11x + 28 = 0\)[/tex] are [tex]\(x = -4\)[/tex] and [tex]\(x = -7\)[/tex].
5. Calculate the corresponding [tex]\( y \)[/tex] values:
For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = (x + 4)^2 - 1 \][/tex]
[tex]\[ y = (-4 + 4)^2 - 1 \][/tex]
[tex]\[ y = 0^2 - 1 \][/tex]
[tex]\[ y = -1 \][/tex]
For [tex]\( x = -7 \)[/tex]:
[tex]\[ y = (x + 4)^2 - 1 \][/tex]
[tex]\[ y = (-7 + 4)^2 - 1 \][/tex]
[tex]\[ y = (-3)^2 - 1 \][/tex]
[tex]\[ y = 9 - 1 \][/tex]
[tex]\[ y = 8 \][/tex]
Therefore, the points that satisfy both equations are [tex]\((-4, -1)\)[/tex] and [tex]\((-7, 8)\)[/tex].
6. Check the solutions against the given options:
- [tex]\((-7, -4)\)[/tex] and [tex]\((8, -1)\)[/tex] - No, these do not match our solutions.
- [tex]\((-7, 8)\)[/tex] and [tex]\((-4, -1)\)[/tex] - Yes, these match our solutions.
- [tex]\((-4, -7)\)[/tex] and [tex]\((-1, 8)\)[/tex] - No, these do not match our solutions.
- [tex]\((8, -7)\)[/tex] and [tex]\((-1, -4)\)[/tex] - No, these do not match our solutions.
Thus, the correct pair of points that solve the system of equations is:
[tex]\[ \boxed{(-7, 8) \text{ and } (-4, -1)} \][/tex]
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