Math Problem Statement
Solution
It seems you've uploaded a set of exercises from Section 3.2 involving solving systems of equations using Gaussian elimination. I'll go ahead and solve one of the equations for you and explain the steps in detail. Let me start with Exercise 1:
Exercise 1:
The system of equations is:
We will solve this using the Gaussian elimination method:
Step 1: Write the system in augmented matrix form
3 & -2 & | & 7 \\ 2 & 4 & | & 10 \end{pmatrix}$$ ### Step 2: Perform row operations to get a row-echelon form Our goal is to make the left side of the matrix an upper triangular matrix. Start by eliminating the $$x$$-term in the second row. - Multiply row 1 by $$2$$ and row 2 by $$3$$ to align the coefficients of $$x$$: $$\begin{pmatrix} 6 & -4 & | & 14 \\ 6 & 12 & | & 30 \end{pmatrix}$$ - Subtract row 1 from row 2: $$\text{Row 2} = \text{Row 2} - \text{Row 1}$$ $$\begin{pmatrix} 6 & -4 & | & 14 \\ 0 & 16 & | & 16 \end{pmatrix}$$ ### Step 3: Solve for $$y$$ From row 2, we have: $$16y = 16 \implies y = 1$$ ### Step 4: Back-substitute to find $$x$$ Substitute $$y = 1$$ into row 1: $$6x - 4(1) = 14 \implies 6x = 18 \implies x = 3$$ ### Solution: The solution to the system is $$x = 3$$, $$y = 1$$. --- Would you like me to solve another exercise or explain Gaussian elimination further? Here are five related questions to consider: 1. How does Gaussian elimination handle larger systems of equations? 2. What is the difference between Gaussian and Gauss-Jordan elimination? 3. How can we check if a system of equations has no solution or infinitely many solutions using matrices? 4. Can Gaussian elimination be applied to non-linear systems of equations? 5. How do row operations preserve the solution of a system? **Tip:** Always check for consistency in the system after applying Gaussian elimination, as it helps detect any contradictions or special cases (no solutions or infinitely many).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Systems
Gaussian Elimination
Formulas
Augmented Matrix Form
Row Reduction
Theorems
Gaussian Elimination Theorem
Suitable Grade Level
Grades 10-12
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