Math Problem Statement
The matrix given is in reduced echelon form.
1
0
3
0
4
3
0
1
−1
0
5
5
0
0
0
1
3
6
Write the system of equations represented by the matrix. (Use x as your variable and label each x with its corresponding column. Enter x_1 for x1, x_2 for x2, x_3 for x3, x_4 for x4, and x_5 for x5.)
row 1
= 3
row 2
= 5
row 3
= 6
Find the solution, if possible. (If there are infinitely many solutions, express x1, x2, x3, x4, and x5 in terms of the parameters s and t. If there is no solution, enter NO SOLUTION.)
Solution
The matrix represents a system of equations. To write the system, each row corresponds to a linear equation, and each column corresponds to a variable (x₁, x₂, x₃, x₄, x₅).
We are given the following reduced echelon form matrix:
1 & 0 & 3 & 0 & 4 & | & 3 \\ 0 & 1 & -1 & 0 & 5 & | & 5 \\ 0 & 0 & 0 & 1 & 3 & | & 6 \end{pmatrix}$$ This matrix represents the system of equations: ### Step 1: Writing the equations - Row 1: $$1x_1 + 0x_2 + 3x_3 + 0x_4 + 4x_5 = 3$$, which simplifies to: $$x_1 + 3x_3 + 4x_5 = 3$$ - Row 2: $$0x_1 + 1x_2 - 1x_3 + 0x_4 + 5x_5 = 5$$, which simplifies to: $$x_2 - x_3 + 5x_5 = 5$$ - Row 3: $$0x_1 + 0x_2 + 0x_3 + 1x_4 + 3x_5 = 6$$, which simplifies to: $$x_4 + 3x_5 = 6$$ ### Step 2: Finding the solution - **Equation 3**: Solve for $$x_4$$: $$x_4 = 6 - 3x_5$$ - **Equation 2**: Solve for $$x_2$$: $$x_2 = 5 + x_3 - 5x_5$$ - **Equation 1**: Solve for $$x_1$$: $$x_1 = 3 - 3x_3 - 4x_5$$ ### Conclusion: Since $$x_3$$ and $$x_5$$ are free variables (they can take any value), the system has **infinitely many solutions**. We can express the solution as: $$x_1 = 3 - 3x_3 - 4x_5$$ $$x_2 = 5 + x_3 - 5x_5$$ $$x_4 = 6 - 3x_5$$ $$x_3 = s \quad (\text{where } s \text{ is any real number})$$ $$x_5 = t \quad (\text{where } t \text{ is any real number})$$ Thus, the solution is: $$x_1 = 3 - 3s - 4t, \quad x_2 = 5 + s - 5t, \quad x_3 = s, \quad x_4 = 6 - 3t, \quad x_5 = t$$ ### Summary: There are infinitely many solutions, and they are expressed in terms of the parameters $$s$$ and $$t$$. Would you like more details or have any questions? Here are some related questions to explore: 1. How do you convert a matrix into reduced row echelon form? 2. What are the conditions for a system of equations to have infinitely many solutions? 3. How does the concept of free variables affect the solution set of a system? 4. What is the significance of pivot positions in a matrix? 5. Can you explain how the back-substitution method works in solving systems of linear equations? **Tip:** Always check if a matrix is in reduced row echelon form by verifying that each leading coefficient (pivot) is 1 and that all entries above and below it are zero.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Row Reduction
Reduced Row Echelon Form
Free Variables
Formulas
Row reduction matrix form
General solution of systems of equations
Theorems
The Rank-Nullity Theorem
Existence and Uniqueness Theorem for Linear Systems
Suitable Grade Level
Undergraduate Level (Introductory Linear Algebra)
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