Math Problem Statement

Express the system of linear equations in the form AX = b.

Solution

The problem asks you to rewrite the system of linear equations in the form AX=bAX = b, where AA is the coefficient matrix, XX is the column vector of variables, and bb is the column vector of constants.

The system of equations is:

x+2y+3z=1x + 2y + 3z = 1 3x+2y+z=23x + 2y + z = 2 2x+y+3z=42x + y + 3z = 4

Step 1: Write the coefficient matrix AA

The coefficients of xx, yy, and zz in each equation are arranged into a matrix:

\begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$$ ### Step 2: Write the variable vector $$X$$ The variables $$x$$, $$y$$, and $$z$$ are written as a column vector: $$X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$ ### Step 3: Write the constant vector $$b$$ The constants on the right-hand side of each equation are written as a column vector: $$b = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$$ ### Final Expression in Matrix Form: $$AX = b$$ Substituting $$A$$, $$X$$, and $$b$$, we get: $$\begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$$ Let me know if you need further clarification or help solving this system! --- Here are 5 related questions to expand your understanding: 1. How do you determine if the system of equations has a unique solution, infinitely many solutions, or no solution? 2. How can you solve the system of equations using Gaussian elimination? 3. What is the determinant of the coefficient matrix $$A$$, and how does it relate to the solution of the system? 4. How would you write the inverse of $$A$$, if it exists, to solve for $$X$$? 5. Can this system be solved graphically, and what does the solution represent geometrically? **Tip:** When writing systems of linear equations in matrix form, double-check each coefficient and constant to ensure no errors during setup!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Representation
Systems of Equations

Formulas

AX = b, where A is the coefficient matrix, X is the variable vector, and b is the constant vector

Theorems

Matrix Representation of Linear Systems

Suitable Grade Level

Grades 10-12

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