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Which of the following represents all solutions (x,y) to the system of equations created by the linear equation and the quadratic equation y = -x2 + 9 ?
A. (3, 0) and (-3, 0)
B. (0, 3) and (0, -3)
C. (2, 5) and (-3, 0)
D. (-2, 1) and (3, 6)
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Given y = x + 3 and y = -x2 + 9 as the linear and quadratic equation respectively.
Since both equations are represented in the same cordinates (x and y), their solution(s) is simply the pioint where both equations intersect.
Thus,
x + 3 = - x2 + 9
rearrange the above equation
x2 + x - 6 = 0
The above is a quadratic equation and can be simplified by factorization
x2 -2x + 3x - 6 = 0
(x2 -2x) + (3x - 6) = 0
x(x -2) + 3(x - 2) = 0
(x + 3) (x - 2) = 0
x + 3 = 0 and x - 2 = 0
x = -3 and x = 2
To get the y-cordinate, we substitute the values of x into one of the two equations.
Substituting into the linear equation yields:
y = -3 + 3 and y = 2 + 3
y = 0 and y = 5
Therefore, the solutions are:
(2, 5) and (-3, 0)
Answer: C
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