Math Problem Statement
solve using elimination
Solve the given system of linear equations using Gaussian elimination.
Solution
This system of linear equations corresponds to the matrix equation:
1 & 0 & -3 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 4 & -4 & -1 \\ 1 & 3 & 0 & -2 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$$ We can solve this system using Gaussian elimination (or row reduction). I'll walk through the elimination process: ### Step 1: Write the augmented matrix. $$\begin{pmatrix} 1 & 0 & -3 & 0 & | & 0 \\ 0 & 1 & -1 & 0 & | & 0 \\ 1 & 4 & -4 & -1 & | & 0 \\ 1 & 3 & 0 & -2 & | & 0 \end{pmatrix}$$ ### Step 2: Perform row operations to reduce the matrix. **Subtract row 1 from row 3 and row 4 to eliminate the first element in rows 3 and 4:** - Row 3: $$R3 - R1$$ $$\begin{pmatrix} 1 & 4 & -4 & -1 & | & 0 \end{pmatrix} - \begin{pmatrix} 1 & 0 & -3 & 0 & | & 0 \end{pmatrix} = \begin{pmatrix} 0 & 4 & -1 & -1 & | & 0 \end{pmatrix}$$ - Row 4: $$R4 - R1$$ $$\begin{pmatrix} 1 & 3 & 0 & -2 & | & 0 \end{pmatrix} - \begin{pmatrix} 1 & 0 & -3 & 0 & | & 0 \end{pmatrix} = \begin{pmatrix} 0 & 3 & 3 & -2 & | & 0 \end{pmatrix}$$ The matrix now looks like this: $$\begin{pmatrix} 1 & 0 & -3 & 0 & | & 0 \\ 0 & 1 & -1 & 0 & | & 0 \\ 0 & 4 & -1 & -1 & | & 0 \\ 0 & 3 & 3 & -2 & | & 0 \end{pmatrix}$$ ### Step 3: Eliminate the second column in row 4 by subtracting an appropriate multiple of row 2. - Row 4: $$R4 - 3 \times R2$$ $$\begin{pmatrix} 0 & 3 & 3 & -2 & | & 0 \end{pmatrix} - 3 \times \begin{pmatrix} 0 & 1 & -1 & 0 & | & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 6 & -2 & | & 0 \end{pmatrix}$$ The matrix is now: $$\begin{pmatrix} 1 & 0 & -3 & 0 & | & 0 \\ 0 & 1 & -1 & 0 & | & 0 \\ 0 & 4 & -1 & -1 & | & 0 \\ 0 & 0 & 6 & -2 & | & 0 \end{pmatrix}$$ ### Step 4: Eliminate the third column in row 3 by subtracting $$\frac{2}{3} \times R4$$ from row 3 to make row 4 have a leading 1: I will continue solving the matrix throughAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Linear Equations
Formulas
Augmented matrix form
Row reduction (Gaussian elimination)
Theorems
Gaussian Elimination Theorem
Linear Independence
Suitable Grade Level
Undergraduate level or advanced high school (Grades 11-12)
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