Math Problem Statement

Consider the system of three linear equations in three variables: 2x + 2z = 2, 5x + 3y = 4, and 3y - 4z = 4. Write a matrix equation for the system in the form A · X = B. Use technology to determine A_inv.

Solution

It looks like you have uploaded a document with two images, which both contain a set of linear equations. Based on the equations shown, you're looking to express the system of equations in matrix form, and possibly find the inverse matrix.

Let's first look at the system of linear equations:

  1. 2x+2z=22x + 2z = 2
  2. 5x+3y=45x + 3y = 4
  3. 3y4z=43y - 4z = 4

To write these in matrix form, we would rewrite the system as:

AX=BA \cdot X = B

Where:

  • AA is the coefficient matrix.
  • XX is the column matrix of variables (i.e., [x,y,z]T[x, y, z]^T).
  • BB is the constants matrix on the right-hand side of the equations.

So, we would extract the coefficients of each equation and write:

A=[202530034],X=[xyz],B=[244]A = \begin{bmatrix} 2 & 0 & 2 \\ 5 & 3 & 0 \\ 0 & 3 & -4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 2 \\ 4 \\ 4 \end{bmatrix}

Thus, the matrix equation is:

[202530034][xyz]=[244]\begin{bmatrix} 2 & 0 & 2 \\ 5 & 3 & 0 \\ 0 & 3 & -4 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 4 \end{bmatrix}

For part b, you would use technology (like a calculator or software) to compute the inverse of matrix AA. The matrix A1A^{-1} would then be used to solve for XX by multiplying both sides of the equation by A1A^{-1}, such that:

X=A1BX = A^{-1} \cdot B

Would you like help with calculating the inverse of matrix AA or any other step?

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Math Problem Analysis

Mathematical Concepts

Matrix Equations
Inverse of a Matrix
Linear Systems
Matrix Multiplication

Formulas

Matrix equation A · X = B
Inverse matrix X = A_inv · B

Theorems

Matrix multiplication results in the identity matrix

Suitable Grade Level

Grades 11-12