Answer:
Explanation:
1. Start with the equation y = -3x - 16.
2. Substitute this expression for y in the other equation.
-x(-3x - 16) = -3x - 16
3. Simplify and solve for x.
3x^2 + 16x = -3x - 16
3x^2 + 16x + 3x + 16 = 0
3x^2 + 19x + 16 = 0
This is a quadratic equation.
4. Factor or use the quadratic formula to solve for x.
Factoring is not straightforward in this case, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = 19, and c = 16.
x = (-19 ± √(19^2 - 4(3)(16))) / (2(3))
x = (-19 ± √(361 - 192)) / 6
x = (-19 ± √169) / 6
x = (-19 ± 13) / 6
5. Simplify further.
x = -32 / 6 or x = -6 / 6
x = -16/3 or x = -1
6. Substitute these values of x back into either equation to find the corresponding values of y.
For x = -16/3:
y = -3(-16/3) - 16
y = 48/3 - 16
y = 16 - 16
y = 0
So, one solution is (x, y) = (-16/3, 0).
For x = -1:
y = -3(-1) - 16
y = 3 - 16
y = -13
The other solution is (x, y) = (-1, -13).
Therefore, the solutions to the system of equations y = -x and y = -3x - 16 are (-16/3, 0) and (-1, -13).