Math Problem Statement
Solution
To solve the system of equations given in the image:
we will use the Gauss-Jordan elimination method by writing the system as an augmented matrix and solving it step by step.
Step 1: Write the system as an augmented matrix
The system can be written in matrix form as follows:
\begin{array}{ccc|c} -5 & -1 & 0 & 10 \\ 10 & -3 & 0 & -45 \end{array} \right]$$ ### Step 2: Perform row operations to solve for $$x$$ and $$y$$ We will now use row operations to transform the matrix into reduced row-echelon form (RREF). 1. **Make the first element of the first row equal to 1** by dividing the first row by $$-5$$: $$R1 \to \frac{R1}{-5}$$ Resulting in: $$\left[ \begin{array}{ccc|c} 1 & \frac{1}{5} & 0 & -2 \\ 10 & -3 & 0 & -45 \end{array} \right]$$ 2. **Eliminate the first element in the second row** by subtracting 10 times the first row from the second row: $$R2 \to R2 - 10R1$$ Resulting in: $$\left[ \begin{array}{ccc|c} 1 & \frac{1}{5} & 0 & -2 \\ 0 & -5 & 0 & -25 \end{array} \right]$$ 3. **Make the second element of the second row equal to 1** by dividing the second row by $$-5$$: $$R2 \to \frac{R2}{-5}$$ Resulting in: $$\left[ \begin{array}{ccc|c} 1 & \frac{1}{5} & 0 & -2 \\ 0 & 1 & 0 & 5 \end{array} \right]$$ 4. **Eliminate the second element in the first row** by subtracting $$\frac{1}{5}$$ times the second row from the first row: $$R1 \to R1 - \frac{1}{5} R2$$ Resulting in: $$\left[ \begin{array}{ccc|c} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 5 \end{array} \right]$$ Now the system is in reduced row-echelon form, meaning we have: $$x = -3 \quad \text{and} \quad y = 5$$ ### Final Solution: The solution to the system of equations is: $$(x, y) = (-3, 5)$$ Would you like further clarification on any steps? Here are some related questions to explore: 1. How is Gauss-Jordan elimination different from Gaussian elimination? 2. What are the possible outcomes when solving a system of linear equations using matrices? 3. Can you explain the significance of row-echelon form and reduced row-echelon form in linear algebra? 4. How would the augmented matrix change if the system had no solution? 5. What happens when you perform row operations that lead to a zero row in the matrix? **Tip**: When using Gauss-Jordan elimination, always ensure that your leading coefficients (pivots) become 1 as you proceed with row operations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Operations
Gauss-Jordan Elimination
Formulas
Row operations (interchange rows, multiply a row by a nonzero scalar, add/subtract multiples of rows)
Reduced Row Echelon Form (RREF)
Theorems
Gauss-Jordan Elimination Theorem
Consistency of Linear Systems
Suitable Grade Level
Grades 11-12, Undergraduate Mathematics
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