1. **State the problem:** We have a table showing the number of students by grade level and whether they have pets or not. Some values are missing, and we need to fill them in. 2. **Given data:** - 6th grade: Pets = 26, Total = 37 - 7th grade: Pets = 0, No Pets = unknown, Total = unknown - 8th grade: Pets = 29, No Pets = unknown, Total = unknown - Total Pets = 55, Total No Pets = unknown, Grand Total = 77 3. **Use the formula:** $$\text{Total} = \text{Pets} + \text{No Pets}$$ 4. **Calculate missing values for 6th grade:** $$\text{No Pets}_{6th} = \text{Total}_{6th} - \text{Pets}_{6th} = 37 - 26 = 11$$ 5. **Calculate total Pets for 7th and 8th grades:** $$\text{Pets}_{7th} + \text{Pets}_{8th} = \text{Total Pets} - \text{Pets}_{6th} = 55 - 26 = 29$$ 6. **Given Pets for 7th grade is 0, so Pets for 8th grade must be:** $$\text{Pets}_{8th} = 29$$ 7. **Calculate No Pets for 8th grade:** Let $x$ be No Pets for 8th grade. 8. **Calculate No Pets total:** $$\text{No Pets total} = \text{Grand Total} - \text{Total Pets} = 77 - 55 = 22$$ 9. **Calculate No Pets for 7th grade:** Let $y$ be No Pets for 7th grade. 10. **Sum of No Pets for 6th, 7th, and 8th grades equals total No Pets:** $$11 + y + x = 22$$ 11. **Calculate total students for 7th and 8th grades:** Let $T_7$ and $T_8$ be totals for 7th and 8th grades. 12. **Total students sum:** $$37 + T_7 + T_8 = 77 \Rightarrow T_7 + T_8 = 40$$ 13. **Express totals in terms of Pets and No Pets:** $$T_7 = 0 + y = y$$ $$T_8 = 29 + x$$ 14. **Sum totals:** $$y + (29 + x) = 40 \Rightarrow y + x = 11$$ 15. **From step 10 and 14, solve system:** $$\begin{cases} 11 + y + x = 22 \\ y + x = 11 \end{cases}$$ Both equations are consistent, so no contradiction. 16. **Calculate $y + x = 11$ and $11 + y + x = 22$ confirms $y + x = 11$.** 17. **Choose $y = 5$ and $x = 6$ (any pair summing to 11 works):** - No Pets 7th grade = 5 - No Pets 8th grade = 6 18. **Calculate totals for 7th and 8th grades:** - Total 7th grade = Pets + No Pets = 0 + 5 = 5 - Total 8th grade = Pets + No Pets = 29 + 6 = 35 19. **Final completed table:** | Grade | Pets | No Pets | Total | |-------|------|---------|-------| | 6th | 26 | 11 | 37 | | 7th | 0 | 5 | 5 | | 8th | 29 | 6 | 35 | | Total | 55 | 22 | 77 | **Answer:** The missing values are No Pets 6th grade = 11, No Pets 7th grade = 5, Total 7th grade = 5, No Pets 8th grade = 6, Total 8th grade = 35, and Total No Pets = 22.