1. **State the problem:** We have a square QRPM with sides QR and PM given as expressions in terms of $x$. 2. **Recall properties of a square:** All sides of a square are equal in length. 3. **Set up the equation:** Since QR and PM are sides of the same square, their lengths must be equal: $$3x - 1 = 8x - 1$$ 4. **Solve for $x$:** $$3x - 1 = 8x - 1$$ $$3x - 1 + 1 = 8x - 1 + 1$$ $$3x = 8x$$ $$\cancel{3x} = \cancel{8x}$$ Subtract $3x$ from both sides: $$3x - 3x = 8x - 3x$$ $$0 = 5x$$ Divide both sides by 5: $$\frac{0}{5} = \frac{5x}{5}$$ $$0 = x$$ 5. **Find the length of side MP:** Substitute $x=0$ into $8x - 1$: $$8(0) - 1 = 0 - 1 = -1$$ 6. **Find the length of side RP:** Since QRPM is a square, all sides are equal, so RP equals QR: $$3x - 1 = 3(0) - 1 = -1$$ 7. **Interpretation:** The side lengths are negative, which is not possible for a square. This suggests no positive $x$ satisfies the given conditions for a square. **Final answer:** $$x = 0$$ Length of side MP = -1$$ (not possible) Length of side RP = -1$$ (not possible)