Final answer:
To find the coordinates of points A and B where a line intersects a curve, we solve the equations simultaneously. The coordinates of point A are (-9, -4) and the coordinates of point B are (-3, -1).
Step-by-step explanation:
To find the coordinates of points A and B, we need to first solve the two equations simultaneously. We'll start by rearranging the equation of the curve to get it into the same form as the equation of the line.
The equation 3y² + 7y + 16 = x - x can be simplified to 3y² + 7y + 16 = 0.
Next, we substitute the value of x from the equation of the line into the equation of the curve, which gives us:
3y² + 7y + 16 = 2y - 1
Combining like terms, we get:
3y² + 5y + 17 = 0
Now we can solve this quadratic equation either by factoring or using the quadratic formula. After solving, we find that the solutions for y are -4 and -1. Plugging these values back into the equation of the line, we can find the corresponding x-values for points A and B.
For y = -4, we have:
2(-4) = x + 1
x = -9
Therefore, the coordinates of point A are (-9, -4).
For y = -1, we have:
2(-1) = x + 1
x = -3
Therefore, the coordinates of point B are (-3, -1).
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