1. **State the problem:** Given mappings from numbers on the left to numbers on the right, find the unknown output, especially for inputs 9 and 19. 2. **Analyze the first set:** 4 → 13, 7 → 22, 1 → 4, 9 → ?. Look for a pattern relating input $x$ to output $y$. 3. **Check differences:** - For $x=1$, output is $4$ - For $x=4$, output is $13$ - For $x=7$, output is $22$ Notice the outputs increase by 9 when inputs increase by 3: $$13-4=9,~~22-13=9$$ This suggests a linear relation $y = mx + b$. 4. **Find linear formula:** Using points $(1,4)$ and $(4,13)$: $$m = \frac{13-4}{4-1} = \frac{9}{3} = 3$$ $$y = 3x + b$$ Substitute $x=1, y=4$: $$4 = 3(1) + b \Rightarrow b = 1$$ Thus, the formula is: $$y = 3x + 1$$ 5. **Find output for $x=9$:** $$y = 3(9) + 1 = 27 + 1 = 28$$ 6. **Analyze second set:** 10 → 12, 19 → 30. Check if the formula $y = 3x + 1$ applies: For $x=10$, $y=3(10)+1=31$, but given is 12, so this formula doesn't fit. Try a simpler relation for second set: Check differences: $$30 - 12 = 18$$ $$19 - 10 = 9$$ Difference in output is twice difference in input. Check formula $y = 2x - 8$: For $x=10$, $y=2(10)-8=20-8=12$ For $x=19$, $y=2(19)-8=38-8=30$ This fits perfectly. **Final answers:** - For $x=9$, output is $28$. - For $x=19$, output is $30$.