Math Problem Statement
Rešiti sistem Gausovom, odnosno Gaus-Žordanovom metodom:
x + y + z = 0,
3x - 2y + 2z = 0,
x - 4y = 0.
Solution
The given system of linear equations is:
1. & \quad x + y + z = 0, \\ 2. & \quad 3x - 2y + 2z = 0, \\ 3. & \quad x - 4y = 0. \end{aligned}$$ We are tasked to solve this system using the **Gaussian** or **Gauss-Jordan elimination** method. --- ### Step 1: Write the augmented matrix The augmented matrix of the system is: $$\begin{bmatrix} 1 & 1 & 1 & 0 \\ 3 & -2 & 2 & 0 \\ 1 & -4 & 0 & 0 \end{bmatrix}.$$ --- ### Step 2: Perform row operations (Gaussian elimination) #### (a) Eliminate the first element in the second and third rows. Subtract $$3 \times \text{(Row 1)}$$ from Row 2 and subtract $$1 \times \text{(Row 1)}$$ from Row 3: $$R_2 \to R_2 - 3R_1, \quad R_3 \to R_3 - R_1$$ $$\begin{bmatrix} 1 & 1 & 1 & 0 \\ 0 & -5 & -1 & 0 \\ 0 & -5 & -1 & 0 \end{bmatrix}.$$ #### (b) Simplify Row 3. Notice $$R_3 = R_2$$, so Row 3 becomes zero: $$\begin{bmatrix} 1 & 1 & 1 & 0 \\ 0 & -5 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$ --- ### Step 3: Back-substitution (Gauss-Jordan elimination) Divide $$R_2$$ by $$-5$$ to make the pivot element $$1$$: $$R_2 \to \frac{R_2}{-5}$$ $$\begin{bmatrix} 1 & 1 & 1 & 0 \\ 0 & 1 & \frac{1}{5} & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$ Now eliminate the $$1$$ in Row 1, Column 2 by subtracting $$R_2$$ from $$R_1$$: $$R_1 \to R_1 - R_2$$ $$\begin{bmatrix} 1 & 0 & \frac{4}{5} & 0 \\ 0 & 1 & \frac{1}{5} & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$ --- ### Step 4: Express the solution The matrix represents the system: $$\begin{aligned} x + \frac{4}{5}z &= 0, \\ y + \frac{1}{5}z &= 0. \end{aligned}$$ Thus, we can solve for $$x$$ and $$y$$ in terms of $$z$$: $$\begin{aligned} x &= -\frac{4}{5}z, \\ y &= -\frac{1}{5}z. \end{aligned}$$ Finally, the solution is: $$(x, y, z) = \left(-\frac{4}{5}z, -\frac{1}{5}z, z\right), \quad z \in \mathbb{R}.$$ This means the solution is a line in parametric form. --- Would you like additional clarification or further details? --- ### Related Questions: 1. How does Gaussian elimination differ from Gauss-Jordan elimination? 2. What does it mean when a system has infinitely many solutions? 3. How can parametric solutions be interpreted geometrically? 4. How can this system be solved graphically? 5. Can this method be applied to non-square systems of equations? ### Tip: Always check if the determinant of the coefficient matrix is zero to quickly assess if the system might have infinitely many solutions or no solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Gauss-Jordan Elimination
Formulas
Row operation: Ri → Ri + cRj
Parametric form: x = a + bt, y = c + dt, ...
Theorems
Gaussian elimination theorem
Solution existence and uniqueness theorem for linear systems
Suitable Grade Level
Undergraduate (Linear Algebra Course)
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