Consider the 3x3 grid with algebraic expressions and constants. What is the value of x if the sum of each row is the same? 4-x x+2 2-x 5-x x 5x 3 2 3+x

Steps to Solve

1. Calculate the sum of the first row Add all the expressions of the first row: $$(4-x) + (x+2) + (2-x)$$

2. Simplify the sum of the first row Combine like terms: $$4 - x + x + 2 + 2 - x = 8 - x$$

3. Calculate the sum of the second row Add all the expressions of the second row: $$(5-x) + (x) + (5x)$$

4. Simplify the sum of the second row Combine like terms: $$5 - x + x + 5x = 5 + 5x$$

5. Calculate the sum of the third row Add all the expressions of the third row: $$3 + 2 + (3+x)$$

6. Simplify the sum of the third row Combine like terms: $$3 + 2 + 3 + x = 8 + x$$

7. Set the sum of the first row equal to the sum of the second row $$8 - x = 5 + 5x$$

8. Solve for x Add $x$ to both sides: $$8 = 5 + 6x$$ Subtract 5 from both sides: $$3 = 6x$$ Divide both sides by 6: $$x = \frac{3}{6} = \frac{1}{2}$$

9. Verify the solution Sum of the first row: $8 - x = 8 - \frac{1}{2} = \frac{16}{2} - \frac{1}{2} = \frac{15}{2}$ Sum of the second row: $5 + 5x = 5 + 5(\frac{1}{2}) = 5 + \frac{5}{2} = \frac{10}{2} + \frac{5}{2} = \frac{15}{2}$ Sum of the third row: $8 + x = 8 + \frac{1}{2} = \frac{16}{2} + \frac{1}{2} = \frac{17}{2}$ Since the sum of the third row does not equal the first and second row, we will now solve for x by setting the first row equal to the third row.

10. Set the sum of the first row equal to the sum of the third row $$8 - x = 8 + x$$

11. Solve for x Subtract 8 from both sides: $$-x = x$$ Add $x$ to both sides: $$0 = 2x$$ Divide by 2: $$x = 0$$

12. Verify the solution Sum of the first row: $8 - x = 8 - 0 = 8$ Sum of the second row: $5 + 5x = 5 + 5(0) = 5 + 0 = 5$ Sum of the third row: $8 + x = 8 + 0 = 8$ Since the sum of the second row does not equal the first and third row, we will now solve for x by setting the second row equal to the third row.

13. **Set the sum of the second row equal to the sum of the third row $$5 + 5x = 8 + x$$

14. Solve for x Subtract 5 from both sides: $$5x = 3 + x$$ Subtract x from both sides: $$4x = 3$$ Divide by 4: $$x = \frac{3}{4}$$

15. Verify the solution Sum of the first row: $8 - x = 8 - \frac{3}{4} = \frac{32}{4} - \frac{3}{4} = \frac{29}{4}$ Sum of the second row: $5 + 5x = 5 + 5(\frac{3}{4}) = 5 + \frac{15}{4} = \frac{20}{4} + \frac{15}{4} = \frac{35}{4}$ Sum of the third row: $8 + x = 8 + \frac{3}{4} = \frac{32}{4} + \frac{3}{4} = \frac{35}{4}$ Since the sum of the first row does not equal the second and third row, we can conclude that none of the rows are equal to each other, therefore, there is no solution.