12th Grade
How to Represent a System of Linear Equations as a Matrix Equation
en_12_alg_matrix_equa_vector_variable(12.06).json
DetailsActivityApprovals
{
"voice_prompt": "Pause briefly between single letters and variables like A and b.",
"manuscript": {
"title": {
"text": "How to Represent a System of Linear Equations as a Matrix Equation",
"audio": "How to Represent a System of Linear Equations as a Matrix Equation"
},
"description": {
"text": "Using matrices, you can represent an entire system of linear equations as a single, compact equation: $A\\mathbf{x} = \\mathbf{b}$. This method organizes the coefficients, variables, and constants into their own matrices and vectors.",
"audio": "Using matrices, you can represent an entire system of linear equations as a single, compact equation: A, times x equals b. This method organizes the coefficients, variables, and constants into their own matrices and vectors."
},
"scenes": [
{
"text": "Before you work with systems of equations, it's helpful to understand what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. You can think of it as a grid or table of numbers.",
"latex": "\\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix}"
},
{
"text": "Matrices are useful because they help you organize and simplify complex problems. They are used in computer graphics, economics, and engineering. In this case, they help you write and solve systems of linear equations more compactly.",
"latex": "a+b"
},
{
"text": "Consider a typical system of two linear equations with two variables, x 1 and x 2. The goal is to rewrite this system as a single matrix equation, A, times x equals b.",
"latex": "a_{11}x_1 + a_{12}x_2 = b_1 \\\\ a_{21}x_1 + a_{22}x_2 = b_2"
},
{
"text": "First, you identify the three main parts from the system. One, the coefficients of the variables. Two, the variables themselves. And three, the constants on the right-hand side of the equations.",
"latex": "a+c"
},
{
"text": "The coefficients are arranged in the 'coefficient matrix', denoted by A. Each row in A, corresponds to an equation, and each column corresponds to a variable. For this system, A is a 2-by-2 matrix.",
"latex": "A = \\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix}"
},
{
"text": "Next, the variables are placed in a column vector, often denoted by x. For this system, x is a column vector with elements x 1 and x 2.",
"latex": "\\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix}"
},
{
"text": "Finally, the constants from the right-hand side of the equations form another column vector, denoted by b. Here, b is a column vector with elements b 1 and b 2.",
"latex": "\\mathbf{b} = \\begin{bmatrix} b_1 \\\\ b_2 \\end{bmatrix}"
},
{
"text": "Now you assemble these parts into the single matrix equation A times x equals b. This compact equation represents the entire original system.",
"latex": "\\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} = \\begin{bmatrix} b_1 \\\\ b_2 \\end{bmatrix}"
},
{
"text": "You can verify that this is correct by performing the matrix-vector multiplication A, times x. The result is a vector whose elements are the left-hand sides of the original equations.",
"latex": "\\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} = \\begin{bmatrix} a_{11}x_1 + a_{12}x_2 \\\\ a_{21}x_1 + a_{22}x_2 \\end{bmatrix}"
},
{
"text": "For a concrete example, take the system 2 x 1 plus 5 x 2 equals 12, and x 1 minus 3 x 2 equals negative 1. The coefficient matrix A is two, five, one, negative three. The variable vector x is x 1, x 2. And the constant vector b, is twelve, negative one.",
"latex": "A = \\begin{bmatrix} 2 & 5 \\\\ 1 & -3 \\end{bmatrix}, \\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix}, \\mathbf{b} = \\begin{bmatrix} 12 \\\\ -1 \\end{bmatrix}"
},
{
"text": "The final matrix equation becomes A, times x equals b.",
"latex": "\\begin{bmatrix} 2 & 5 \\\\ 1 & -3 \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} = \\begin{bmatrix} 12 \\\\ -1 \\end{bmatrix}"
}
],
"outro": {
"text": "You can rewrite entire systems of equations into the compact matrix form A times x equals b. This is a fundamental step for using powerful matrix algebra to find solutions.",
"audio": "You can rewrite entire systems of equations into the compact matrix form A, times x equals b. This is a fundamental step for using powerful matrix algebra to find solutions."
}
}
}