Math Problem Statement
{−2x−15y=9−2x−14y=4 Use elementary row operations on the augmented matrix in part (a) to make a matrix of the form
Solution
We are given the system of linear equations:
-2x - 15y &= 9 \\ -2x - 14y &= 4 \end{aligned}$$ The goal is to use elementary row operations to transform the augmented matrix into a simplified form. First, let’s represent the system as an augmented matrix. ### Step 1: Write the Augmented Matrix The augmented matrix corresponding to the system is: $$\begin{pmatrix} -2 & -15 & 9 \\ -2 & -14 & 4 \end{pmatrix}$$ ### Step 2: Apply Row Operations To simplify the matrix, we will perform elementary row operations. Here’s the plan: 1. Make the coefficient of $$x$$ in the first row equal to 1. 2. Use that row to eliminate the $$x$$-coefficient from the second row. 3. Continue simplifying until the matrix is in row-echelon form (the desired matrix form). #### Step 2.1: Row 1 - Divide by -2 Divide Row 1 by -2 to make the first element a 1: $$R1 \rightarrow \frac{R1}{-2}$$ This gives: $$\begin{pmatrix} 1 & \frac{15}{2} & -\frac{9}{2} \\ -2 & -14 & 4 \end{pmatrix}$$ #### Step 2.2: Row 2 - Add Row 1 multiplied by 2 to Row 2 Now, eliminate the $$x$$-term from the second row by performing the operation: $$R2 \rightarrow R2 + 2 \cdot R1$$ $$R2 = \begin{pmatrix} -2 & -14 & 4 \end{pmatrix} + 2 \cdot \begin{pmatrix} 1 & \frac{15}{2} & -\frac{9}{2} \end{pmatrix}$$ Calculating this, we get: $$R2 = \begin{pmatrix} 0 & 1 & -5 \end{pmatrix}$$ Thus, the matrix becomes: $$\begin{pmatrix} 1 & \frac{15}{2} & -\frac{9}{2} \\ 0 & 1 & -5 \end{pmatrix}$$ ### Step 3: Simplify Now that the matrix is in an upper triangular form (row-echelon form), this is the desired result. The matrix represents the simplified system: $$\begin{aligned} x + \frac{15}{2} y &= -\frac{9}{2} \\ y &= -5 \end{aligned}$$ ### Step 4: Solve the System From the second equation, we can directly see that $$y = -5$$. Substituting $$y = -5$$ into the first equation: $$x + \frac{15}{2}(-5) = -\frac{9}{2}$$ $$x - \frac{75}{2} = -\frac{9}{2}$$ $$x = -\frac{9}{2} + \frac{75}{2} = \frac{66}{2} = 33$$ So, the solution is $$x = 33$$ and $$y = -5$$. ### Conclusion: The system has a unique solution: $$x = 33$$ and $$y = -5$$. Would you like further details on any of the steps? Here are five related questions to consider: 1. How can you apply Gaussian elimination to larger systems of equations? 2. What are the different types of elementary row operations? 3. Can you solve a system if the matrix has no unique solution? 4. How can you use row-reduction to find the inverse of a matrix? 5. What is the difference between row-echelon form and reduced row-echelon form? **Tip:** Always check the determinant of a matrix when solving systems—it can indicate whether the system has a unique 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
Row-Echelon Form
Formulas
Row operations: Row replacement, row scaling, row swapping
Solution of a system of linear equations using augmented matrices
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 10-12