Math Problem Statement

Write the system of equations represented by the matrix in reduced echelon form, then find the solution in terms of a parameter if there are infinitely many solutions.

Solution

Let's analyze the matrix in reduced echelon form:

1 & 0 & 2 & -1 \\ 0 & 1 & 5 & 4 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ Each row represents an equation. Let's write the system of equations: ### Step 1: Write the Equations 1. **Row 1**: $$1x_1 + 0x_2 + 2x_3 = -1$$ - This simplifies to: $$x_1 + 2x_3 = -1$$. 2. **Row 2**: $$0x_1 + 1x_2 + 5x_3 = 4$$ - This simplifies to: $$x_2 + 5x_3 = 4$$. 3. **Row 3**: $$0 = 0$$, which is always true and doesn’t add any new information. So the system of equations is: $$x_1 + 2x_3 = -1$$ $$x_2 + 5x_3 = 4$$ ### Step 2: Express in Terms of Free Variable Since $$x_3$$ is not constrained by any leading variable (it has no pivot in any row), it’s a free variable. Let $$x_3 = t$$, where $$t$$ is any real number. Now we can express $$x_1$$ and $$x_2$$ in terms of $$t$$: 1. From $$x_1 + 2x_3 = -1$$: $$x_1 = -1 - 2t$$ 2. From $$x_2 + 5x_3 = 4$$: $$x_2 = 4 - 5t$$ ### Solution The solution in terms of the parameter $$t$$ is: $$(x_1, x_2, x_3) = (-1 - 2t, 4 - 5t, t)$$ Would you like more details or have any questions? ### Related Questions: 1. How can we determine if a system has infinitely many solutions? 2. What are the characteristics of a matrix in reduced echelon form? 3. How do free variables affect the solution set of a system? 4. What is the difference between row echelon form and reduced row echelon form? 5. How can this solution be interpreted geometrically? #### Tip: For a matrix to represent a consistent system, any row of zeros must correspond to a true statement like $$0 = 0$$.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Systems of Linear Equations
Reduced Row Echelon Form

Formulas

Row operations for reduced echelon form

Theorems

Matrix Row Reduction
Free Variable Theorem

Suitable Grade Level

Grades 11-12 or College Algebra

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