Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. { x + 3 y + 4 z = − 3 x + 2 y + 3 z = − 2 x + 4 y + 3 z = − 6 \begin{cases}x + 3y + 4z = -3 \\x + 2y + 3z = -2 \\x + 4y + 3z = -6\end{cases} ⎩ ⎨ ⎧ x + 3 y + 4 z = − 3 x + 2 y + 3 z = − 2 x + 4 y + 3 z = − 6

express the following system of equations. As a matrix equation where we have this set of three equations. Now, a matrix equation is generally given by the form A times X equals B or a star coefficient matrix B is our constant matrix. Excellent. Br variable matrix. So let's go and take a look. We want to find what A. Is. Now. They will be the coefficients of everything on the left side of our equal side. If you notice here we have two, five and 9 on the top row, those are the coefficients. Our next one would be 1 -3, 7 And finally four Native three. This is our matrix A. Now multiply environ matrix X which serve variable matrix well we know we want X, Y and Z as our variables. So we will say X, Y. C. This is equal to our constant matrix. The constant matrix is just the matrix of everything. On the right side of the equal side. We notice here we have negative five negative one and negative seven. We can't simplify this any further. So this is our matrix equation right here meaning the answer to our problem that's actually answer A. Okay, I hope that you solve the problem. Thank you for watching. Goodbye

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

1:50 minutes

Problem 31

Textbook Question

Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.
$\begin{cases}\)x + 3y + 4z = -3 \$\x$ + 2y + 3z = -2 \$\x$ + 4y + 3z = -6\(\end{cases}$

Verified step by step guidance

Identify the variables and write them as a column matrix $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$.

Extract the coefficients of the variables from each equation to form the coefficient matrix $A = \begin{bmatrix} 1 & 3 & 4 \\ 1 & 2 & 3 \\ 1 & 4 & 3 \end{bmatrix}$.

Write the constants from the right side of each equation as the constant matrix $B = \begin{bmatrix} -3 \\ -2 \\ -6 \end{bmatrix}$.

Express the system of equations in the matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix.

The final matrix equation is $\begin{bmatrix} 1 & 3 & 4 \\ 1 & 2 & 3 \\ 1 & 4 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -3 \\ -2 \\ -6 \end{bmatrix}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and solve these systems is fundamental in algebra.

Matrix Representation of Linear Systems

A system of linear equations can be expressed in matrix form as AX = B, where A is the coefficient matrix containing the coefficients of variables, X is the column matrix of variables, and B is the constant matrix. This form simplifies solving and analyzing the system.

Coefficient and Constant Matrices

The coefficient matrix (A) includes all coefficients of the variables arranged by equation and variable order, while the constant matrix (B) contains the constants from the right side of each equation. Correctly identifying these matrices is essential for forming the matrix equation.

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