Constructing Augmented Matrix for System of Linear Equations

Math Problem Statement

Construct the augmented matrix that corresponds to the following system of equations: (7 + 6x)/7 = y, 7z - 4(x - 5y) = 0, 5x - y = 2(x - 3z)

To construct the augmented matrix corresponding to the system of equations in the image, let’s break down the given system:

We'll simplify each equation.

$\frac{7 + 6x}{7} = y \quad \Rightarrow \quad 7 + 6x = 7y$ Thus, the equation becomes: $6x - 7y = -7$

$7z - 4(x - 5y) = 0 \quad \Rightarrow \quad 7z - 4x + 20y = 0$ Thus, the equation becomes: $-4x + 20y + 7z = 0$

$5x - y = 2(x - 3z) \quad \Rightarrow \quad 5x - y = 2x - 6z$ Thus, the equation becomes: $3x - y + 6z = 0$

The system of equations now is:

The augmented matrix is:

6 & -7 & 0 & | & -7 \\ -4 & 20 & 7 & | & 0 \\ 3 & -1 & 6 & | & 0 \end{pmatrix}$$ Let me know if you'd like more details on any step! ### Further questions you could explore: 1. How do you perform row reduction to solve the augmented matrix? 2. What is the geometric interpretation of the solutions to this system? 3. How can you solve a system of equations using matrix inversion? 4. What is the difference between homogeneous and non-homogeneous systems of equations? 5. How would the system change if one of the equations was inconsistent? ### Tip: When constructing augmented matrices, make sure that the coefficients are aligned correctly and remember to include the constants in the last column.

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Linear Algebra
Systems of Equations
Augmented Matrix

Augmented matrix representation of systems of equations
Algebraic simplification

Gaussian Elimination
Matrix Representation of Linear Systems

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