Math Problem Statement
Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, set
x3 = t
and solve for x1 and x2.)
x1
−
3x3
=
−5
3x1
+
x2
−
2x3
=
1
2x1
+
2x2
+
x3
=
1
(x1, x2, x3) =
Solution
To solve the system, we can use Gaussian elimination with back-substitution. Let’s rewrite the system in matrix form:
x_1 - 3x_3 = -5 \\ 3x_1 + x_2 - 2x_3 = 1 \\ 2x_1 + 2x_2 + x_3 = 1 \end{cases}$$ ### Step 1: Write the Augmented Matrix The augmented matrix for this system is: $$\begin{bmatrix} 1 & 0 & -3 & | & -5 \\ 3 & 1 & -2 & | & 1 \\ 2 & 2 & 1 & | & 1 \\ \end{bmatrix}$$ ### Step 2: Make the First Column into Echelon Form 1. Use row 1 as the pivot. Eliminate the $$3x_1$$ in row 2 and the $$2x_1$$ in row 3. - Row 2: $$R_2 - 3 \times R_1$$: $$\begin{bmatrix} 0 & 1 & 7 & | & 16 \\ \end{bmatrix}$$ - RowAsk a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Gauss-Jordan Elimination
Formulas
Gaussian Elimination steps for solving systems of equations
Theorems
Gaussian Elimination Method
Gauss-Jordan Elimination Method
Suitable Grade Level
Grades 10-12