1. **Problem Statement:** Determine the geometry of the systems of equations given by (a) and (b) using normal analysis. 2. **Recall:** The geometry of three planes in 3D can be: - Three planes intersecting at a single point. - Two planes coincident and the third intersecting them. - Three planes intersecting along a line. - No common intersection (parallel or inconsistent). 3. **System (a):** \begin{align*} 3x - 5y + 2z &= 7 \\ 4x + y - z &= 6 \\ 15x - 25y + 10z &= 35 \end{align*} Note that the third equation is exactly 5 times the first: $$15x - 25y + 10z = 5(3x - 5y + 2z) = 5 \times 7 = 35$$ This means the third plane coincides with the first plane. 4. **Reduced system:** Only two distinct planes: \begin{align*} 3x - 5y + 2z &= 7 \\ 4x + y - z &= 6 \end{align*} 5. **Find the line of intersection of these two planes:** We solve for $x,y,z$ satisfying both equations. From the first: $$3x - 5y + 2z = 7$$ From the second: $$4x + y - z = 6$$ Express $y$ and $z$ in terms of $x$ or parametrize. Multiply second equation by 2: $$8x + 2y - 2z = 12$$ Add to first equation: $$(3x - 5y + 2z) + (8x + 2y - 2z) = 7 + 12$$ $$11x - 3y = 19$$ Express $y$: $$-3y = 19 - 11x \implies y = \frac{11x - 19}{3}$$ From second equation: $$4x + y - z = 6 \implies z = 4x + y - 6$$ Substitute $y$: $$z = 4x + \frac{11x - 19}{3} - 6 = \frac{12x + 11x - 19 - 18}{3} = \frac{23x - 37}{3}$$ 6. **Parametric form of line of intersection:** Let $x = t$, then $$y = \frac{11t - 19}{3}, \quad z = \frac{23t - 37}{3}$$ 7. **Geometry for (a):** Two distinct planes intersecting in a line; the third plane coincides with the first. --- 8. **System (b):** \begin{align*} 2x - y - z &= 5 \\ 3x + 4y + z &= -4 \\ 9x + y - 2z &= 6 \end{align*} 9. **Check if planes intersect at a point:** Form coefficient matrix and check rank: $$A = \begin{bmatrix}2 & -1 & -1 \\ 3 & 4 & 1 \\ 9 & 1 & -2\end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix}5 \\ -4 \\ 6\end{bmatrix}$$ Calculate determinant of $A$: $$\det(A) = 2(4 \times -2 - 1 \times 1) - (-1)(3 \times -2 - 1 \times 9) + (-1)(3 \times 1 - 4 \times 9)$$ $$= 2(-8 -1) + 1(-6 -9) -1(3 -36) = 2(-9) + 1(-15) -1(-33) = -18 -15 +33 = 0$$ Determinant is zero, so planes do not intersect at a single point. 10. **Check ranks:** Rank of $A$ and augmented matrix $[A|b]$ to determine consistency. Augmented matrix: $$\begin{bmatrix}2 & -1 & -1 & 5 \\ 3 & 4 & 1 & -4 \\ 9 & 1 & -2 & 6\end{bmatrix}$$ Perform row operations to check rank: - Multiply row 1 by 3 and row 2 by 2 and subtract: $$3R1: (6, -3, -3, 15), 2R2: (6, 8, 2, -8)$$ $$2R2 - 3R1 = (0, 11, 5, -23)$$ - Multiply row 1 by 9 and row 3 by 2 and subtract: $$9R1: (18, -9, -9, 45), 2R3: (18, 2, -4, 12)$$ $$2R3 - 9R1 = (0, 11, 5, -33)$$ Rows 2 and 3 after operation: $$R2': (0, 11, 5, -23), R3': (0, 11, 5, -33)$$ Since the last elements differ, rows are inconsistent. 11. **Conclusion for (b):** The system is inconsistent; the three planes do not intersect at a common point or line. **Final answers:** - (a) Two distinct planes intersecting in a line; third plane coincides with one. - (b) No common intersection; planes are inconsistent.