Math Problem Statement
Solve the following system using an augmented matrix. If this is dependent, write using parameters. Write as a matrix and also as a sum of vectors. x1 – 2x2 + x4 = 3 -x1 + x2 – x3 – 8x4 = 1
Solution
To solve the given system of equations using an augmented matrix, let's first rewrite the system clearly:
Step 1: Represent the system as an augmented matrix.
We will express the coefficients of the variables and the constants in an augmented matrix form:
1 & -2 & 0 & 1 & | & 3 \\ -1 & 1 & -1 & -8 & | & 1 \end{pmatrix}$$ ### Step 2: Row reduction (Gaussian elimination). We aim to simplify this matrix into row-echelon form using elementary row operations. The matrix is: $$\begin{pmatrix} 1 & -2 & 0 & 1 & | & 3 \\ -1 & 1 & -1 & -8 & | & 1 \end{pmatrix}$$ - Add row 1 to row 2 to eliminate the first element in row 2: $$R_2 = R_2 + R_1$$ $$\begin{pmatrix} 1 & -2 & 0 & 1 & | & 3 \\ 0 & -1 & -1 & -7 & | & 4 \end{pmatrix}$$ - Now, multiply row 2 by $$-1$$ to simplify: $$R_2 = -R_2$$ $$\begin{pmatrix} 1 & -2 & 0 & 1 & | & 3 \\ 0 & 1 & 1 & 7 & | & -4 \end{pmatrix}$$ ### Step 3: Back substitution. Now that the matrix is in row-echelon form, we can convert it back to equations: 1. From the first row: $$x_1 - 2x_2 + x_4 = 3$$ 2. From the second row: $$x_2 + x_3 + 7x_4 = -4$$ ### Step 4: Solving the system. We can express $$x_1$$ and $$x_2$$ in terms of free variables $$x_3$$ and $$x_4$$. From the second equation: $$x_2 = -4 - x_3 - 7x_4$$ Substitute this into the first equation: $$x_1 - 2(-4 - x_3 - 7x_4) + x_4 = 3$$ $$x_1 + 8 + 2x_3 + 14x_4 + x_4 = 3$$ $$x_1 = 3 - 8 - 2x_3 - 15x_4$$ $$x_1 = -5 - 2x_3 - 15x_4$$ ### Step 5: Write the solution using parameters. Since $$x_3$$ and $$x_4$$ are free variables, we can parametrize the solution: $$x_1 = -5 - 2x_3 - 15x_4$$ $$x_2 = -4 - x_3 - 7x_4$$ $$x_3 = x_3 \quad \text{(free)}$$ $$x_4 = x_4 \quad \text{(free)}$$ ### Step 6: Write the solution as a matrix and as a sum of vectors. The parametric solution can be written as: $$\begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} -5 \\ -4 \\ 0 \\ 0 \end{pmatrix} + x_3 \begin{pmatrix} -2 \\ -1 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} -15 \\ -7 \\ 0 \\ 1 \end{pmatrix}$$ This represents the solution as a linear combination of vectors. --- Would you like further details on any step? Here are some related questions: 1. How do elementary row operations affect the solution set of a system? 2. How can we tell if a system is dependent by looking at its augmented matrix? 3. What is the geometric interpretation of the solution as a sum of vectors? 4. Can this method be generalized to larger systems? 5. How does the number of free variables relate to the rank of a matrix? **Tip:** When dealing with systems of equations, free variables correspond to the number of degrees of freedom in the solution, giving insight into whether the system has infinite solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Augmented Matrix
Vector Representation
Formulas
Row-reduction (Gaussian Elimination)
Parametric Solution Form
Linear Combination of Vectors
Theorems
Gaussian Elimination
Free Variables in Systems of Equations
Suitable Grade Level
Grades 10-12 and Early College
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