Math Problem Statement
Find the solution of the following system using Gauss elimination. (Enter your answers as a comma-separated list.)
x
−
y
−
z
+
4u
=
-10
x
−
y
+
z
+
4u
=
-6
y
+
2z
−
u
=
8
4y
+
z
+
2u
=
12
(x, y, z, u) = (
Solution
We are given the following system of equations:
x - y - z + 4u &= -10 \quad \text{(Equation 1)} \\ x - y + z + 4u &= -6 \quad \text{(Equation 2)} \\ y + 2z - u &= 8 \quad \text{(Equation 3)} \\ 4y + z + 2u &= 12 \quad \text{(Equation 4)} \end{align*}$$ We will use Gaussian elimination to solve this system of equations. Let's write the augmented matrix for the system: $$\begin{pmatrix} 1 & -1 & -1 & 4 & -10 \\ 1 & -1 & 1 & 4 & -6 \\ 0 & 1 & 2 & -1 & 8 \\ 0 & 4 & 1 & 2 & 12 \end{pmatrix}$$ ### Step 1: Eliminate $$x$$-terms from rows 2, 3, and 4 We subtract row 1 from row 2 to eliminate $$x$$-term in row 2: $$R_2 = R_2 - R_1 \implies \begin{pmatrix} 1 & -1 & -1 & 4 & -10 \\ 0 & 0 & 2 & 0 & 4 \\ 0 & 1 & 2 & -1 & 8 \\ 0 & 4 & 1 & 2 & 12 \end{pmatrix}$$ ### Step 2: Eliminate $$y$$-terms from rows 3 and 4 Next, we subtract 4 times row 3 from row 4 to eliminate the $$y$$-term from row 4: $$R_4 = R_4 - 4R_3 \implies \begin{pmatrix} 1 & -1 & -1 & 4 & -10 \\ 0 & 0 & 2 & 0 & 4 \\ 0 & 1 & 2 & -1 & 8 \\ 0 & 0 & -7 & 6 & -20 \end{pmatrix}$$ ### Step 3: Back-substitution Now, we can start solving the system from the last row. - From row 4: $$-7z + 6u = -20$$ Solving for $$z$$: $$z = \frac{6u + 20}{7}$$ We substitute this expression for $$z$$ into row 3. - From row 3: $$y + 2z - u = 8$$ Substituting $$z = \frac{6u + 20}{7}$$ into the equation: $$y + 2\left(\frac{6u + 20}{7}\right) - u = 8$$ Simplifying, we solve for $$y$$ and then substitute these into the other equations. I'll compute the exact values for $$x, y, z, u$$.The solution to the system of equations is: $$(x, y, z, u) = (-1, 3, 2, -1)$$ Would you like further details on the steps, or do you have any questions? Here are some related questions to consider: 1. How does Gaussian elimination work for non-square systems? 2. What is the significance of pivoting in Gaussian elimination? 3. Can Gaussian elimination always provide a unique solution? 4. How do row operations affect the determinant of a matrix? 5. What are some methods for solving systems of equations with more unknowns than equations? **Tip:** Gaussian elimination is most efficient when dealing with systems that have a unique solution, but for systems with infinitely many solutions, it can be combined with parameterization techniques.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Linear Equations
Formulas
Gaussian Elimination steps
Row operations (row addition, subtraction, scaling)
Back-substitution
Theorems
Gaussian Elimination Theorem
Unique Solutions in Linear Systems
Suitable Grade Level
Grades 10-12
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