Write each matrix equation as a system of linear equations without matrices. [ 4 − 7 2 − 3 ] [ x y ] = [ − 3 1 ] \begin{bmatrix}4 & -7 \\2 & -3\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}-3 \\1\end{bmatrix} [ 4 2 − 7 − 3 ] [ x y ] = [ − 3 1 ]

express the following matrix equation. As a system of equations. We have this set of matrices right here to do this, we're gonna multiply our matrices here. We have eight. Native to 56. Multiplied by X. Y. Multiply this makeshift together. The first part tells us we're gonna multiply our row top row by our color. So in this case we have eight x first. Now we go 1, 2 or -2. They have two times where that's equal to the part on the right, negative left. Now we can do the bottom row are bottom row was 56. We will get five times x Plus six times y equals four. Now we can't simplify this any further. So if you notice the answer to our problem is actually answer D. Which is this set of equations? Okay. I hope to help 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

1:12 minutes

Problem 33

Textbook Question

Write each matrix equation as a system of linear equations without matrices.
$\begin{bmatrix}\)4 & -7 \\2 & -3$\end{bmatrix}\[\begin{bmatrix}$x \$\y\]\end{bmatrix}$=$\begin{bmatrix}$-3 \\1\(\end{bmatrix}$

Verified step by step guidance

Identify the given matrix equation: $\left[ \begin{array}{cc} 4 & -7 \\ 2 & -3 \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} -3 \\ 1 \end{array} \right]$.

Recall that multiplying a matrix by a vector corresponds to forming linear combinations of the vector components with the matrix rows.

Write the first row multiplication as an equation: \$4x - 7y = -3$.

Write the second row multiplication as an equation: \$2x - 3y = 1$.

Thus, the matrix equation is equivalent to the system of linear equations: $\begin{cases} 4x - 7y = -3 \\ 2x - 3y = 1 \end{cases}$.

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Key Concepts

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

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. In this problem, multiplying the 2x2 coefficient matrix by the 2x1 variable matrix results in a 2x1 matrix representing the system's left side.

System of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Writing the matrix equation as a system means expressing each row multiplication as an individual linear equation involving variables x and y.

Equating Matrices to Form Equations

When two matrices are equal, their corresponding entries are equal. This principle allows us to set each element of the product matrix equal to the corresponding element in the constant matrix, forming a system of equations to solve.

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Master Determinants of 3×3 Matrices with a bite sized video explanation from Patrick

Write each matrix equation as a system of linear equations without matrices.[4−72−3][xy]=[−31]\(\begin{bmatrix}\)4 & -7 \\2 & -3\(\end{bmatrix}\[\begin{bmatrix}\)x \\(\y\]\end{bmatrix}\)=\(\begin{bmatrix}\)-3 \\1\(\end{bmatrix}\)[42−7−3][xy]=[−31]

Key Concepts

Matrix Multiplication

System of Linear Equations

Equating Matrices to Form Equations

Watch next

Write each matrix equation as a system of linear equations without matrices.
$\begin{bmatrix}\)4 & -7 \\2 & -3\(\end{bmatrix}\[\begin{bmatrix}\)x \\(\y\]\end{bmatrix}\)=\(\begin{bmatrix}\)-3 \\1\(\end{bmatrix}$