Solving a System of Linear Equations with Free Parameter Using Matrix Algebra

Math Problem Statement

eq1 = Eq(3x + 12y - 13z, 4) eq2 = Eq(2x + 9y - 10z, 0) eq3 = Eq(x + 3y - 3z, 4)

To solve the given system of linear equations:

We will solve it step by step using a systematic approach.

The system can be written as $AX = B$ , where:

\begin{bmatrix} 3 & 12 & -13 \\ 2 & 9 & -10 \\ 1 & 3 & -3 \end{bmatrix}, \; X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \; B = \begin{bmatrix} 4 \\ 0 \\ 4 \end{bmatrix}.$$ ### Step 2: Solve the system Let me calculate the solution. sympy import symbols, Eq, solve, Matrix # Define variables x, y, z = symbols('x y z') # Define the equations eq1 = Eq(3*x + 12*y - 13*z, 4) eq2 = Eq(2*x + 9*y - 10*z, 0) eq3 = Eq(x + 3*y - 3*z, 4) # Solve the system of equations solution = solve((eq1, eq2, eq3), (x, y, z)) solutionThe solution to the system of equations is: $$x = 12 - z$$ $$y = \frac{4z}{3} - \frac{8}{3}$$ $$z = z \quad (\text{free parameter})$$ This implies that the system has infinitely many solutions, with $$z$$ acting as a free parameter. You can substitute any value for $$z$$ to find corresponding values for $$x$$ and $$y$$. Would you like detailed examples of how specific $$z$$ values affect the solution or further clarification on solving such systems? ### Expanding Understanding: 1. How does a free parameter in a solution relate to the geometry of the system? 2. What does the rank of the coefficient matrix tell us about the system's solutions? 3. How would the solution change if an additional equation were added? 4. What is the physical interpretation of a parameterized solution in real-world applications? 5. How can we verify the solution for consistency? **Tip:** Always check the rank of the coefficient matrix to confirm the number of free parameters in the solution!

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Linear Algebra
Systems of Linear Equations
Matrix Algebra

AX = B (Matrix form of a system of equations)
Matrix inversion (when applicable)
Row-reduction to echelon form

Rank-Nullity Theorem
System Consistency Theorem

Grades 10-12 or College Level

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