1. **Stating the problem:** We are given several algebraic expressions and equations involving $x$ and $y$, and we want to find the value of $x$ and check the correctness of the steps. 2. **Given equations:** - $\frac{x}{4} + \frac{yx^2}{2x^2} = 5$ - $\frac{x + 2y}{4} = \frac{5}{1}$ - $x + 2(2x - 4) = 20$ - $\frac{x x^2}{2x^2} - \frac{y}{4} = 1$ - $x + 2y = 20$ - $x + 4x - 8 = 20$ - $5x - 8 = 20$ - $\frac{2x - y}{4} = 1$ 3. **Step-by-step analysis:** **Step 1:** Simplify $\frac{yx^2}{2x^2}$ in the first equation: $$\frac{yx^2}{2x^2} = \frac{y \cancel{x^2}}{2 \cancel{x^2}} = \frac{y}{2}$$ So the first equation becomes: $$\frac{x}{4} + \frac{y}{2} = 5$$ **Step 2:** Multiply both sides by 4 to clear denominators: $$4 \times \left(\frac{x}{4} + \frac{y}{2}\right) = 4 \times 5$$ $$x + 2y = 20$$ This matches the equation $x + 2y = 20$ given later. **Step 3:** From the equation $x + 2(2x - 4) = 20$, expand the parentheses: $$x + 4x - 8 = 20$$ $$5x - 8 = 20$$ Add 8 to both sides: $$5x - 8 + 8 = 20 + 8$$ $$5x = 28$$ Divide both sides by 5: $$\frac{5x}{5} = \frac{28}{5}$$ $$x = \frac{28}{5} = 5.6$$ **Step 4:** From $\frac{2x - y}{4} = 1$, multiply both sides by 4: $$2x - y = 4$$ Add $y$ and subtract 4 on both sides to isolate $y$: $$2x - 4 = y$$ **Step 5:** Substitute $y = 2x - 4$ into $x + 2y = 20$: $$x + 2(2x - 4) = 20$$ $$x + 4x - 8 = 20$$ $$5x - 8 = 20$$ This is consistent with Step 3. **Step 6:** The value of $x$ found is $\frac{28}{5}$ or 5.6. **Where was the mistake?** - The main error is in the last line where it says $5x/5 = 28/5$ but then does not simplify correctly or write the final value of $x$. - Also, the original expression $\frac{x x^2}{2x^2}$ should be simplified carefully: $$\frac{x x^2}{2x^2} = \frac{x^3}{2x^2} = \frac{\cancel{x^2} x}{2 \cancel{x^2}} = \frac{x}{2}$$ - The user wrote $\frac{x x^2}{2x^2} - \frac{y}{4} = 1$ but did not simplify it fully. **Summary:** The algebraic manipulations are mostly correct except for missing simplifications and final answers. The key is to simplify fractions carefully and write the final values explicitly. **Final answer:** $$x = \frac{28}{5} = 5.6$$ $$y = 2x - 4 = 2 \times \frac{28}{5} - 4 = \frac{56}{5} - 4 = \frac{56}{5} - \frac{20}{5} = \frac{36}{5} = 7.2$$