Question: 7. Solve the following non-linear system of equations. {\ny = x^2 - x - 20\n3x + y = -17\n
1. **State the problem:** We need to solve the system of equations:
$$y = x^2 - x - 20$$
$$3x + y = -17$$
2. **Use substitution method:** Since $y$ is expressed in terms of $x$ in the first equation, substitute $y$ from the first equation into the second:
$$3x + (x^2 - x - 20) = -17$$
3. **Simplify the equation:**
$$3x + x^2 - x - 20 = -17$$
$$x^2 + (3x - x) - 20 = -17$$
$$x^2 + 2x - 20 = -17$$
4. **Bring all terms to one side:**
$$x^2 + 2x - 20 + 17 = 0$$
$$x^2 + 2x - 3 = 0$$
5. **Factor the quadratic equation:**
$$x^2 + 2x - 3 = (x + 3)(x - 1) = 0$$
6. **Solve for $x$:**
$$x + 3 = 0 \Rightarrow x = -3$$
$$x - 1 = 0 \Rightarrow x = 1$$
7. **Find corresponding $y$ values:**
- For $x = -3$:
$$y = (-3)^2 - (-3) - 20 = 9 + 3 - 20 = -8$$
- For $x = 1$:
$$y = 1^2 - 1 - 20 = 1 - 1 - 20 = -20$$
8. **Final solution:**
$$\boxed{(-3, -8) \text{ and } (1, -20)}$$