Math Problem Statement
Solve using Gauss-Jordan elimination.
33x 1x1minus−
7 x 27x2
minus−
5 x 3 equals5x3=
2626
x 1x1minus−
4 x 24x2
equals=
1212
Question content area bottom
Part 1
Select the correct choice below and fill in the answer box(es) within your choice.
A.
The unique solution is
x 1x1equals=enter your response here,
x 2x2equals=enter your response here,
and
x 3x3equals=enter your response here.
B.
The system has infinitely many solutions. The solution is
x 1x1equals=enter your response here,
x 2x2equals=enter your response here,
and
x 3x3equals=t.
(Simplify your answers. Type expressions using t as the variable.)
C.
The system has infinitely many solutions. The solution is
x 1x1equals=enter your response here,
x 2x2equals=s,
and
x 3x3equals=t.
(Simplify your answer. Type an expression using s and t as the variables.)
D.
There is no solution.
Solution
Let's first organize the given system of equations and solve it using Gauss-Jordan elimination.
Given System of Equations:
Step 1: Write the augmented matrix
We write the augmented matrix for the system of equations:
33 & -7 & -5 & | & 26 \\ 1 & -4 & 0 & | & 12 \end{pmatrix}$$ ### Step 2: Perform row operations to reduce to row-echelon form Our goal is to reduce the matrix to the form where we can easily read off the solutions. We will apply Gauss-Jordan elimination. 1. We already have a 1 in the second row, first column. We'll use that to eliminate the first entry of the first row. Subtract $$33 \times \text{(row 2)}$$ from row 1: $$R_1 \rightarrow R_1 - 33R_2$$ Perform the calculation: $$(33, -7, -5 | 26) - 33 \times (1, -4, 0 | 12)$$ $$= (33, -7, -5 | 26) - (33, -132, 0 | 396)$$ $$= (0, 125, -5 | -370)$$ So the augmented matrix now looks like: $$\begin{pmatrix} 0 & 125 & -5 & | & -370 \\ 1 & -4 & 0 & | & 12 \end{pmatrix}$$ ### Step 3: Make the coefficient of $$x_2$$ in row 1 a 1 Divide the first row by 125: $$R_1 \rightarrow \frac{R_1}{125}$$ $$\left( 0, 125, -5 | -370 \right) \rightarrow \left( 0, 1, -\frac{1}{25} | -\frac{74}{25} \right)$$ The new augmented matrix is: $$\begin{pmatrix} 0 & 1 & -\frac{1}{25} & | & -\frac{74}{25} \\ 1 & -4 & 0 & | & 12 \end{pmatrix}$$ ### Step 4: Eliminate the $$x_2$$ term from the second row Add $$4 \times \text{(row 1)}$$ to row 2: $$R_2 \rightarrow R_2 + 4R_1$$ Perform the calculation: $$(1, -4, 0 | 12) + 4 \times (0, 1, -\frac{1}{25} | -\frac{74}{25})$$ $$= (1, -4, 0 | 12) + (0, 4, -\frac{4}{25} | -\frac{296}{25})$$ $$= (1, 0, -\frac{4}{25} | \frac{4}{25})$$ So the augmented matrix is now: $$\begin{pmatrix} 0 & 1 & -\frac{1}{25} & | & -\frac{74}{25} \\ 1 & 0 & -\frac{4}{25} & | & \frac{4}{25} \end{pmatrix}$$ ### Step 5: Solve for the variables From row 1: $$x_2 - \frac{1}{25}x_3 = -\frac{74}{25}$$ $$x_2 = -\frac{74}{25} + \frac{1}{25}x_3$$ From row 2: $$x_1 - \frac{4}{25}x_3 = \frac{4}{25}$$ $$x_1 = \frac{4}{25} + \frac{4}{25}x_3$$ ### Step 6: Conclusion Since $$x_3$$ is free (it can take any value), the system has infinitely many solutions. The solution is: $$x_1 = \frac{4}{25} + \frac{4}{25}t, \quad x_2 = -\frac{74}{25} + \frac{1}{25}t, \quad x_3 = t$$ where $$t$$ is a free variable. ### Final Answer: The correct choice is **B**: $$x_1 = \frac{4}{25} + \frac{4}{25}t, \quad x_2 = -\frac{74}{25} + \frac{1}{25}t, \quad x_3 = t$$ --- Would you like more details or have any questions? Here are five related problems to try: 1. Solve a 3x3 system of linear equations using Gauss-Jordan elimination. 2. What are the advantages of Gauss-Jordan elimination over Gaussian elimination? 3. How do free variables impact the solution set of a system of linear equations? 4. Can every system of linear equations be solved using Gauss-Jordan elimination? 5. What is the geometric interpretation of a system of linear equations with infinitely many solutions? **Tip**: Always check if your matrix is in reduced row-echelon form before concluding the solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gauss-Jordan Elimination
Formulas
Row reduction using Gauss-Jordan elimination
Augmented matrix representation of systems of equations
Theorems
Gauss-Jordan elimination method
Row-Echelon Form
Suitable Grade Level
College Algebra or Higher
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