Math Problem Statement
User responses cleared Homework:12.2 HW - Systems of Linear Equations: Matrices Question 20, 12.2.47 HW Score: 52.14%, 18.25 of 35 points Points: 0 of 1
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5 x minus 2 y equals5 10 x plus 2 y equals13 Question content area bottom Part 1 Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The solution is left parenthesis nothing comma nothing right parenthesis . (Simplify your answers.) B. There are infinitely many solutions. The solution can be written as StartSet left parenthesis x comma y right parenthesis font size decreased by 6 | font size decreased by 6 x equals nothing comma y is any real number EndSet . (Simplify your answer. Type an expression using y as the variable.) C. The system is inconsistent. If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.If the system has no solution, say that it is inconsistent.Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. Ask my instructor
Solution
To solve this system of equations using matrices and row operations, let’s start by writing it in matrix form:
The system is:
Step 1: Write the Augmented Matrix
The augmented matrix for this system is:
5 & -2 & | & 5 \\ 10 & 2 & | & 13 \\ \end{pmatrix}$$ ### Step 2: Perform Row Operations 1. **Make the leading coefficient of the first row equal to 1** by dividing the first row by 5: $$\begin{pmatrix} 1 & -\frac{2}{5} & | & 1 \\ 10 & 2 & | & 13 \\ \end{pmatrix}$$ 2. **Eliminate the x-term in the second row** by subtracting 10 times the first row from the second row: $$\text{Row 2} = \text{Row 2} - 10 \times \text{Row 1}$$ $$\begin{pmatrix} 1 & -\frac{2}{5} & | & 1 \\ 0 & 6 & | & 3 \\ \end{pmatrix}$$ 3. **Make the leading coefficient of the second row equal to 1** by dividing the second row by 6: $$\begin{pmatrix} 1 & -\frac{2}{5} & | & 1 \\ 0 & 1 & | & \frac{1}{2} \\ \end{pmatrix}$$ 4. **Eliminate the y-term in the first row** by adding $$\frac{2}{5}$$ times the second row to the first row: $$\text{Row 1} = \text{Row 1} + \frac{2}{5} \times \text{Row 2}$$ $$\begin{pmatrix} 1 & 0 & | & \frac{9}{5} \\ 0 & 1 & | & \frac{1}{2} \\ \end{pmatrix}$$ ### Step 3: Interpret the Solution The final matrix corresponds to the following equations: $$x = \frac{9}{5}$$ $$y = \frac{1}{2}$$ ### Conclusion The solution to the system is: $$\left( x, y \right) = \left( \frac{9}{5}, \frac{1}{2} \right)$$ ### Answer The correct choice is: **A. The solution is** $$\left( \frac{9}{5}, \frac{1}{2} \right)$$. Would you like additional details on any step, or have any questions? Here are some related questions to explore further: 1. How do we determine if a system of equations has infinitely many solutions using matrices? 2. What row operations can we perform to solve a system of equations in matrix form? 3. How do we identify an inconsistent system from an augmented matrix? 4. What is the difference between Gaussian elimination and Gauss-Jordan elimination? 5. How do we check the solution to a system of linear equations? **Tip:** Always ensure each leading coefficient is 1 in row echelon form for clearer interpretation of the solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Row Operations
Formulas
Row operations to achieve row echelon form
Matrix representation of linear systems
Theorems
Row Echelon Form
Gaussian Elimination
Gauss-Jordan Elimination
Suitable Grade Level
Grades 10-12
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