1. **State the problem:** Given the table with values of $x$ and $y$, find the equation relating $x$ and $y$ and then find the missing $y$ value when $x=20$. 2. **Analyze the data:** The table is: $$\begin{array}{c|ccccc} x & 1 & 4 & 8 & 12 & 20 \\ y & 4 & 19 & 39 & 59 & ? \\\end{array}$$ 3. **Check if the relationship is linear:** Calculate the differences in $y$ values: $19 - 4 = 15$ $39 - 19 = 20$ $59 - 39 = 20$ The differences are not constant, so the relation is not perfectly linear. 4. **Try to find a pattern or fit a quadratic equation:** Assume the equation is of the form: $$y = ax^2 + bx + c$$ 5. **Use three points to form equations:** For $x=1, y=4$: $$a(1)^2 + b(1) + c = 4 \Rightarrow a + b + c = 4$$ For $x=4, y=19$: $$a(4)^2 + b(4) + c = 19 \Rightarrow 16a + 4b + c = 19$$ For $x=8, y=39$: $$a(8)^2 + b(8) + c = 39 \Rightarrow 64a + 8b + c = 39$$ 6. **Solve the system:** From equation 1: $c = 4 - a - b$ Substitute into equations 2 and 3: $$16a + 4b + (4 - a - b) = 19 \Rightarrow 15a + 3b + 4 = 19$$ $$64a + 8b + (4 - a - b) = 39 \Rightarrow 63a + 7b + 4 = 39$$ Simplify: $$15a + 3b = 15$$ $$63a + 7b = 35$$ 7. **Divide first equation by 3:** $$5a + b = 5$$ Divide second equation by 7: $$9a + b = 5$$ 8. **Subtract the two equations:** $$(9a + b) - (5a + b) = 5 - 5 \Rightarrow 4a = 0 \Rightarrow a = 0$$ 9. **Find $b$:** $$5(0) + b = 5 \Rightarrow b = 5$$ 10. **Find $c$:** $$c = 4 - 0 - 5 = -1$$ 11. **Equation is:** $$y = 0 \cdot x^2 + 5x - 1 = 5x - 1$$ 12. **Check with $x=12$:** $$y = 5(12) - 1 = 60 - 1 = 59$$ which matches the table. 13. **Find $y$ when $x=20$:** $$y = 5(20) - 1 = 100 - 1 = 99$$ **Final answer:** The equation is $y = 5x - 1$ and the missing $y$ value is $99$.