Math Problem Statement
Assignment#3.pdf
597.18 KB
Solve question 5
Solution
The system of equations in Question 5 is:
x + 2y - 2z &= 5 \\ x - y + 3z &= -1 \\ x - 7y + kz &= -k \end{aligned}$$ ### (a) Reduce the system to echelon form: We will use Gaussian elimination to reduce the system to row echelon form. The augmented matrix of the system is: $$\begin{pmatrix} 1 & 2 & -2 & | & 5 \\ 1 & -1 & 3 & | & -1 \\ 1 & -7 & k & | & -k \end{pmatrix}$$ **Step 1:** Subtract row 1 from rows 2 and 3 to eliminate $$x$$ from rows 2 and 3. $$R2 = R2 - R1 \quad \text{and} \quad R3 = R3 - R1$$ This gives: $$\begin{pmatrix} 1 & 2 & -2 & | & 5 \\ 0 & -3 & 5 & | & -6 \\ 0 & -9 & k+2 & | & -k-5 \end{pmatrix}$$ **Step 2:** Divide row 2 by -3 to simplify: $$R2 = \frac{R2}{-3}$$ Now the matrix becomes: $$\begin{pmatrix} 1 & 2 & -2 & | & 5 \\ 0 & 1 & -\frac{5}{3} & | & 2 \\ 0 & -9 & k+2 & | & -k-5 \end{pmatrix}$$ **Step 3:** Use row 2 to eliminate $$y$$ from row 3. We add 9 times row 2 to row 3: $$R3 = R3 + 9R2$$ This gives: $$\begin{pmatrix} 1 & 2 & -2 & | & 5 \\ 0 & 1 & -\frac{5}{3} & | & 2 \\ 0 & 0 & k - 1 & | & -k + 13 \end{pmatrix}$$ This is the echelon form of the system. ### (b) Infinitely many solutions: For the system to have infinitely many solutions, the last row must represent a true statement (i.e., 0 = 0). This happens when the coefficient of $$z$$ in the last row is zero, and the constant on the right-hand side is also zero. So, set: $$k - 1 = 0 \quad \text{and} \quad -k + 13 = 0$$ From $$k - 1 = 0$$, we get $$k = 1$$. Substitute $$k = 1$$ into $$-k + 13 = 0$$: $$-1 + 13 = 12 \neq 0$$ There was an error above. Only the first condition, $$k - 1 = 0$$, needs to be checked. ThisAsk a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Row Echelon Form
Gaussian Elimination
Formulas
Row operations for Gaussian Elimination
Condition for infinite solutions (k = 1)
Theorems
Gaussian Elimination Theorem
Consistent Systems Theorem
Suitable Grade Level
Undergraduate
Related Recommendation
Solve for h and k in a Linear System using Gaussian Elimination
Find k for System of Linear Equations with Infinitely Many Solutions
Gaussian Elimination: Finding General and Particular Solutions for a System of Linear Equations
Solve a System of Linear Equations Using Gaussian Elimination
Solving a System of Linear Equations Using Gaussian Elimination