1. The problem asks to find the best approximate solutions of the system of equations where $f(x)$ and $g(x)$ intersect. 2. The intersection points occur where $f(x) = g(x)$. 3. From the table, we compare values of $f(x)$ and $g(x)$ for each $x$: | $x$ | $f(x)$ | $g(x)$ | |------|-------|-------| | 1.10 | 0.49 | 0.23 | | 1.15 | 0.32 | 0.22 | | 1.20 | 0.14 | 0.20 | | 1.25 | 0.04 | 0.18 | | 1.30 | 0.21 | 0.17 | | 1.35 | 0.39 | 0.15 | | 1.40 | 0.56 | 0.13 | 4. We look for $x$ values where $f(x)$ and $g(x)$ are closest. 5. At $x=1.25$, $f(x)=0.04$ and $g(x)=0.18$, difference $=|0.04-0.18|=0.14$. 6. At $x=1.30$, $f(x)=0.21$ and $g(x)=0.17$, difference $=|0.21-0.17|=0.04$. 7. At $x=1.35$, $f(x)=0.39$ and $g(x)=0.15$, difference $=|0.39-0.15|=0.24$. 8. The smallest difference is at $x=1.30$ with difference $0.04$, so the best approximate solution is near $x=1.30$. 9. The approximate solution point is $(1.30, 0.21)$ or $(1.30, 0.17)$ since $f(x)$ and $g(x)$ values are close. 10. Therefore, the best approximate solutions of the system are near $x=1.30$ with $y$ values approximately $0.21$ and $0.17$. Final answer: The approximate solutions of the system of equations exist where $f(x)$ and $g(x)$ intersect. The best approximate solutions of the system of equations are near $x=1.30$ and $y=0.21$ or $y=0.17$.