Math Problem Statement
A system of equations is rewritten as a matrix, which is then converted to row echelon form. Which of the following can be concluded from the matrix in its row echelon form?
Solution
The matrix given in row echelon form is:
1 & 1 & 1 & 0 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 4 \end{pmatrix}$$ This matrix represents the following system of equations: 1. $$x + y + z = 0$$ (from the first row) 2. $$y + 2z = 3$$ (from the second row) 3. $$z = 4$$ (from the third row) ### Solving step-by-step: - From the third row, we already know $$z = 4$$. - Substituting $$z = 4$$ into the second row equation $$y + 2z = 3$$: $$y + 2(4) = 3 \implies y + 8 = 3 \implies y = -5$$ - Now substitute $$y = -5$$ and $$z = 4$$ into the first equation $$x + y + z = 0$$: $$x + (-5) + 4 = 0 \implies x - 1 = 0 \implies x = 1$$ Thus, the solution to the system is $$x = 1$$, $$y = -5$$, and $$z = 4$$. ### Correct conclusions from the matrix: - From the third row: $$z = 4$$ (Option A is correct). - From the second row: $$y + 2z = 3$$ (Option E is correct). ### Answer: The correct answers are **A (z = 4)** and **E (y + 2z = 3)**. --- Would you like more details or have any questions? Here are 5 related questions for further practice: 1. How do you determine if a system is consistent or inconsistent from a row-echelon matrix? 2. Can a system of equations have no solution if the matrix is in row echelon form? Why or why not? 3. What is the significance of leading ones in row echelon form? 4. How does row reduction help in solving a system of linear equations? 5. How would the solution change if the matrix had a different number of rows or columns? **Tip:** Always check the consistency of a system by looking at the last row in the echelon form for a possible contradiction (like 0 = non-zero).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Systems
Matrices
Row Echelon Form
Formulas
Matrix row reduction
System of linear equations
Substitution method
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 9-12