Math Problem Statement
Use Gaussian elimination and back-substitution to solve the following system of equations. If there is a solution, write your answer in the format (x, y). -2x - 2y = 16 and -4x - 4y = 31
Solution
The system of equations given in the image is:
-2x - 2y &= 16 \\ -4x - 4y &= 31 \end{aligned}$$ To solve this system using Gaussian elimination: ### Step 1: Set up the augmented matrix $$\begin{pmatrix} -2 & -2 & | & 16 \\ -4 & -4 & | & 31 \end{pmatrix}$$ ### Step 2: Perform row operations 1. The first row can be simplified by dividing by $$-2$$, which gives: $$\begin{pmatrix} 1 & 1 & | & -8 \\ -4 & -4 & | & 31 \end{pmatrix}$$ 2. To eliminate the $$x$$-term from the second row, we can perform the operation: $$R_2 \rightarrow R_2 - (-4) \times R_1$$ Applying this operation results in: $$\begin{pmatrix} 1 & 1 & | & -8 \\ 0 & 0 & | & -1 \end{pmatrix}$$ ### Step 3: Analyze the resulting matrix The second row is $$0 = -1$$, which is a contradiction. This means that the system of equations is inconsistent and has **no solution**. ### Conclusion: The correct answer is **No Solution**. --- Do you want more details or have any questions? Here are five related questions to explore this topic further: 1. How does Gaussian elimination work for systems with infinitely many solutions? 2. What is the difference between consistent and inconsistent systems of equations? 3. Can a system have exactly two solutions? Why or why not? 4. How would you solve a system of three equations using Gaussian elimination? 5. What are the advantages of using Gaussian elimination over substitution? **Tip:** Always check for consistency when using Gaussian elimination. If a row of the form $$0 = c$$ appears, the system is inconsistent and has no solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Equations
Inconsistent Systems
Formulas
Row Reduction
Gaussian Elimination
Matrix Representation
Theorems
Gaussian Elimination Process
Inconsistent System Theorem
Suitable Grade Level
Grades 10-12
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