Solve the following system by the substitution method.
$\begin{array}{l} x + y = 3 \\ y = x^{2} - 17 \end{array}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
(Type an ordered pair. Use a comma to separate answers as needed. Simplify your answer.)
B. There is no solution.

Answer

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Answer

Thus, the solution set is $(2, 1)$ and $(11, - 8)$ .

Steps

Step 1 ：First, we can express $x$ in terms of $y$ from the first equation: $x = 3 - y$ .

Step 2 ：Next, we substitute $x$ into the second equation: $y = (3 - y)^{2} - 17$ .

Step 3 ：This simplifies to: $y = 9 - 6 y + y^{2} - 17$ .

Step 4 ：Rearranging terms, we get a quadratic equation: $y^{2} + 7 y - 8 = 0$ .

Step 5 ：We can solve this quadratic equation by factoring: $(y - 1) (y + 8) = 0$ .

Step 6 ：So, the solutions for $y$ are $y = 1$ and $y = - 8$ .

Step 7 ：For $y = 1$ , we substitute back into the first equation to find $x$ : $x = 3 - 1 = 2$ .

Step 8 ：For $y = - 8$ , we substitute back into the first equation to find $x$ : $x = 3 - (- 8) = 11$ .

Step 9 ：Thus, the solution set is $(2, 1)$ and $(11, - 8)$ .