Question
Gaussian elimination for the system with M and v
Original question: with(Student[LinearAlgebra]) eq1 := 2 x + y + 4 z = 5; eq2 := x + y + 2 z = 4; eq3 := -x - 2 z = -1 M := Matrix([[2, 1, 4], [1, 1, 2], [1, 0, -2]]) v := Vector([5, 4, 1]); GaussianEliminationTutor(M, v)
Expert Verified Solution
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Expert intro: This is a linear algebra system-solving task. The matrix form and the equations describe the same system, so Gaussian elimination is the natural method.
Detailed walkthrough
We solve the system
2x+y+4z&=5\\ x+y+2z&=4\\ -x-2z&=-1 \end{aligned}$$ ### Step 1: Write the augmented matrix $$\left[\begin{array}{ccc|c} 2&1&4&5\\ 1&1&2&4\\ -1&0&-2&-1 \end{array}\right]$$ The matrix from the prompt is $$M=\begin{bmatrix}2&1&4\\1&1&2\\1&0&-2\end{bmatrix}, \quad v=\begin{bmatrix}5\\4\\1\end{bmatrix}$$ which corresponds to the system above, with the third equation multiplied by $-1$. ### Step 2: Use the third equation From $$-x-2z=-1$$ we get $$x+2z=1$$ so $$x=1-2z$$ ### Step 3: Substitute into the second equation $$x+y+2z=4$$ Substitute $x=1-2z$: $$1-2z+y+2z=4$$ $$y=3$$ ### Step 4: Substitute into the first equation $$2x+y+4z=5$$ Substitute $y=3$: $$2x+3+4z=5$$ $$2x+4z=2$$ $$x+2z=1$$ This is consistent with Step 2. So the system has infinitely many solutions, with one free parameter. Let $z=t$. Then $$x=1-2t, \quad y=3, \quad z=t$$ ### Solution set $$\boxed{(x,y,z)=(1-2t,\,3,\,t),\ t\in\mathbb{R}}$$ ### 💡 Pitfall guide Do not assume every linear system has a unique solution. Here the equations are dependent, so elimination leads to a free variable. Also, note that the matrix row $[1,0,-2]$ with vector entry $1$ matches the equation $x-2z=1$, which is equivalent to $-x-2z=-1$ after multiplying by $-1$. ### 🔄 Real-world variant If the third equation were changed to $-x-2z=-3$, then the substitution would give $x=3-2z$. That would change the final solution set completely. If you want, the same system can also be analyzed by row reduction to identify the rank and null space structure. ### 🔍 Related terms augmented matrix, Gaussian elimination, free variable