Math Problem Statement
Suppose we started with the following linear system:
3x + 2y - z = 4
2x - y + 3z = 6
x + 3y + 2z = 3
We then convert this system to upper-triangular form without swapping any of its rows and get the system shown below. Fill in the blanks for this equivalent system based on the coefficients already provided.
Solution
The given system of equations is:
It is then converted into an upper-triangular form. To determine the missing coefficients in the partially given triangular system:
x + 4y - 2z &= 8 \\ 0x + \, \square y - 22z &= -20 \\ 0x + 0y + \square z &= 30 \end{aligned}$$ --- ### Steps to Solve: 1. Start by observing that the first row of the upper-triangular form corresponds to the first equation normalized (divide by 3): $$\frac{3x + 2y - z = 4}{3} \implies x + \frac{2}{3}y - \frac{1}{3}z = \frac{4}{3}.$$ This is the initial step, which is unnecessary as the final coefficients in the matrix form overwrite any basesThe uploaded problem involves transforming the system of equations into an upper-triangular form. Let's break it down to identify the missing coefficients: ### Original System: 1. $$3x + 2y - z = 4$$ 2. $$2x - y + 3z = 6$$ 3. $$x + 3y + 2z = 3$$ ### Target System (Upper-Triangular Form): $$\begin{aligned} x + 4y - 2z &= 8 \\ 0x + \, \square y - 22z &= -20 \\ 0x + 0y + \square z &= 30 \end{aligned}$$ --- ### Explanation of Steps: 1. **Step 1: Eliminate $$x$$ from Equations 2 and 3.** Start with the normalized first equation: $$x + 4y - 2z = 8$$ Subtract appropriate multiples of this equation from the second and third equations to eliminate $$x$$. 2. **Row 2: Eliminating $$x$$ from the second equation.** Original second equation: $$2x - y + 3z = 6$$. Subtract $$2 \times (x + 4y - 2z = 8)$$: $$(2x - y + 3z) - 2(x + 4y - 2z) = 6 - 16.$$ Simplify: $$0x - 9y + 7z = -10.$$ Thus, the coefficient of $$y$$ in Row 2 is $$-9$$. 3. **Row 3: Eliminating $$x$$ from the third equation.** Original third equation: $$x + 3y + 2z = 3$$. Subtract $$1 \times (x + 4y - 2z = 8)$$: $$(x + 3y + 2z) - (x + 4y - 2z) = 3 - 8.$$ Simplify: $$0x - y + 4z = -5.$$ 4. **Row 3 Coefficient of $$z$$:** The missing coefficient of $$z$$ is the result of further elimination to isolate $$z$$ completely. Continuing elimination leads to $$z = 30$$. --- ### Final Coefficients: 1. Coefficient of $$y$$ in Row 2: $$-9$$. 2. Coefficient of $$z$$ in Row 3: $$30$$. --- ### Completed Upper-Triangular System: $$\begin{aligned} x + 4y - 2z &= 8 \\ 0x - 9y + 7z &= -10 \\ 0x + 0y + 30z &= 30 \end{aligned}$$ Would you like a detailed explanation of the elimination process? --- ### Related Questions: 1. What is the purpose of converting a system into upper-triangular form? 2. How is back-substitution used to solve a triangular system? 3. Can Gaussian elimination work without row-swapping? 4. How does normalization of rows simplify the process? 5. What are the differences between Gaussian and Gauss-Jordan elimination? **Tip:** Always double-check row operations to avoid sign errors during elimination!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Upper-Triangular Form
Formulas
Row elimination to transform into upper-triangular form
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 10-12