1. **Stating the problem:** We have a 3x3 grid with arithmetic operations and numbers, some missing values represented by ellipses (...). The goal is to find the missing numbers so that the equations in rows and columns hold true. 2. **Given:** - Row 1: ? + 9 + ... = 13 - Row 2: 7 + ... - 5 = 6 - Row 3: ... - ... + 8 = 12 - Column 1: ? + 7 + ... = 4 - Column 2: 9 + ... - ... = 7 - Column 3: ... - 5 + 8 = 14 3. **Define variables:** Let the missing numbers be $x$, $y$, $z$, $w$, $u$, $v$ corresponding to the ellipses and question mark: - Row 1: $a + 9 + x = 13$ (where $a$ is the question mark) - Row 2: $7 + y - 5 = 6$ - Row 3: $z - w + 8 = 12$ - Column 1: $a + 7 + z = 4$ - Column 2: $9 + y - w = 7$ - Column 3: $x - 5 + 8 = 14$ 4. **Solve equations step-by-step:** From Row 1: $$a + 9 + x = 13 \implies a + x = 4$$ From Row 2: $$7 + y - 5 = 6 \implies y + 2 = 6 \implies y = 4$$ From Row 3: $$z - w + 8 = 12 \implies z - w = 4$$ From Column 1: $$a + 7 + z = 4 \implies a + z = -3$$ From Column 2: $$9 + y - w = 7 \implies 9 + 4 - w = 7 \implies 13 - w = 7 \implies w = 6$$ From Column 3: $$x - 5 + 8 = 14 \implies x + 3 = 14 \implies x = 11$$ 5. **Substitute $x=11$ into $a + x = 4$:** $$a + 11 = 4 \implies a = -7$$ 6. **Substitute $a = -7$ into $a + z = -3$:** $$-7 + z = -3 \implies z = 4$$ 7. **Recall $z - w = 4$ and $w=6$:** $$4 - 6 = -2$$ This contradicts $z - w = 4$. So re-check calculations. 8. **Re-examine Column 1:** Given $a + 7 + z = 4$, with $a = -7$, then: $$-7 + 7 + z = 4 \implies z = 4$$ 9. **Check $z - w = 4$ with $z=4$ and $w=6$:** $$4 - 6 = -2 \neq 4$$ 10. **Contradiction found, re-check Column 2:** $$9 + y - w = 7$$ We had $y=4$, so: $$9 + 4 - w = 7 \implies 13 - w = 7 \implies w = 6$$ 11. **Try to solve $z - w = 4$ with $w=6$:** $$z - 6 = 4 \implies z = 10$$ 12. **Recalculate Column 1 with $a = -7$ and $z=10$:** $$a + 7 + z = 4 \implies -7 + 7 + 10 = 10 \neq 4$$ 13. **Adjust $a$ and $z$ to satisfy both equations:** From Row 1: $$a + x = 4$$ From Column 1: $$a + 7 + z = 4 \implies a + z = -3$$ From Row 3: $$z - w = 4$$ From Column 2: $$9 + y - w = 7$$ From Row 2: $$7 + y - 5 = 6 \implies y = 4$$ From Column 3: $$x + 3 = 14 \implies x = 11$$ 14. **Use $y=4$, $x=11$ and solve for $w$, $z$, $a$:** From Column 2: $$9 + 4 - w = 7 \implies 13 - w = 7 \implies w = 6$$ From Row 3: $$z - 6 = 4 \implies z = 10$$ From Column 1: $$a + 7 + 10 = 4 \implies a + 17 = 4 \implies a = -13$$ From Row 1: $$-13 + 11 = -2 \neq 4$$ 15. **Contradiction again, try to solve $a$ and $x$ from Row 1 and Column 1 simultaneously:** From Row 1: $$a + x = 4$$ From Column 1: $$a + 7 + z = 4 \implies a + z = -3$$ From Row 3: $$z - w = 4$$ From Column 2: $$9 + y - w = 7$$ From Row 2: $$7 + y - 5 = 6 \implies y = 4$$ From Column 3: $$x + 3 = 14 \implies x = 11$$ 16. **Calculate $w$ from Column 2:** $$9 + 4 - w = 7 \implies 13 - w = 7 \implies w = 6$$ 17. **Calculate $z$ from Row 3:** $$z - 6 = 4 \implies z = 10$$ 18. **Calculate $a$ from Column 1:** $$a + 7 + 10 = 4 \implies a + 17 = 4 \implies a = -13$$ 19. **Check Row 1:** $$a + x = -13 + 11 = -2 \neq 4$$ 20. **Since contradiction persists, check if bottom row numbers 4, 7, 14 correspond to column sums:** - Column 1 sum: $a + 7 + z = 4$ - Column 2 sum: $9 + y - w = 7$ - Column 3 sum: $x - 5 + 8 = 14$ 21. **All consistent except Row 1 sum. Possibly Row 1 sum is 13, so:** $$a + 9 + x = 13$$ Substitute $a = -13$, $x=11$: $$-13 + 9 + 11 = 7 \neq 13$$ 22. **Try to solve $a$ and $x$ from Row 1 only:** $$a + 9 + x = 13 \implies a + x = 4$$ 23. **Try $a = -1$, $x = 5$:** - Check Column 3: $$x - 5 + 8 = 14 \implies 5 - 5 + 8 = 8 \neq 14$$ 24. **Try $x=11$ from Column 3, then $a = 4 - 11 = -7$ from Row 1:** - Check Column 1: $$a + 7 + z = 4 \implies -7 + 7 + z = 4 \implies z = 4$$ - Check Row 3: $$z - w + 8 = 12 \implies 4 - w + 8 = 12 \implies 12 - w = 12 \implies w = 0$$ - Check Column 2: $$9 + y - w = 7 \implies 9 + y - 0 = 7 \ ightarrow y = -2$$ - Check Row 2: $$7 + y - 5 = 6 \implies 7 - 2 - 5 = 0 \ eq 6$$ 25. **Try $y=4$ from Row 2:** $$7 + 4 - 5 = 6$$ Correct. 26. **Try $w=6$ from Column 2:** $$9 + 4 - 6 = 7$$ Correct. 27. **Try $z=10$ from Row 3:** $$10 - 6 + 8 = 12$$ Correct. 28. **Try $a = -13$ from Column 1:** $$-13 + 7 + 10 = 4$$ Correct. 29. **Try $x=11$ from Column 3:** $$11 - 5 + 8 = 14$$ Correct. 30. **Check Row 1:** $$-13 + 9 + 11 = 7 \neq 13$$ 31. **Conclusion:** The only inconsistency is Row 1 sum. Possibly a typo or misinterpretation. **Final values:** $$a = -13, x = 11, y = 4, w = 6, z = 10$$ **Answer:** - Missing values are $a = -13$, $x = 11$, $y = 4$, $w = 6$, $z = 10$