Show that the system of linear equations 2 x1+3 x2+x3 =5 x1-4 x2+5 x3 =8 can be writte

Show that the system of linear equations 2 x1+3 x2+x3 =5...

Show that the system of linear equations $$ \begin{aligned} 2 x_{1}+3 x_{2}+x_{3} &=5 \\ x_{1}-4 x_{2}+5 x_{3} &=8 \end{aligned} $$ can be written as the matrix equation $$ \left[\begin{array}{rrr}{2} & {3} & {1} \\ {1} & {-4} & {5}\end{array}\right]\left[\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right]=\left[\begin{array}{l}{5} \\ {8}\end{array}\right] $$

Show that the system of linear equations
$$
\begin{aligned} 2 x_{1}+3 x_{2}+x_{3} &=5 \\ x_{1}-4 x_{2}+5 x_{3} &=8 \end{aligned}
$$
can be written as the matrix equation
$$
\left[\begin{array}{rrr}{2} & {3} & {1} \\ {1} & {-4} & {5}\end{array}\right]\left[\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right]=\left[\begin{array}{l}{5} \\ {8}\end{array}\right]
$$

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Key Concepts

System of Linear Equations

A system of linear equations is a collection of one or more equations involving the same set of variables, where each equation describes a linear relationship between the variables. Solving the system means finding values for the variables that satisfy all the equations simultaneously.

Matrix Representation of Linear Systems

The matrix representation involves expressing a system of linear equations in the compact form Ax = b, where A is the coefficient matrix containing the numerical coefficients of the variables, x is the column vector of the unknowns, and b is the column vector of the constants. This formulation simplifies manipulation, analysis, and computational methods in linear algebra.

Matrix Multiplication

Matrix multiplication is the process by which the coefficient matrix A is multiplied by the variable vector x to produce the resulting vector that corresponds to the linear combinations dictated by the system’s equations. This operation encapsulates the entire system in one equation, making it easier to apply various solution techniques and understand the structure of the system.

Transcript

00:01 For this problem, we have a system of linear equations in three variables, x sub 1, x sub 2, and x sub 3.

00:08 And we want to compare this system of linear equations to a matrix multiplication equation.

00:15 And that equation looks like this, 2, 3, 1, 1, negative 4, 5.

00:23 And we're going to multiply that by x sub 1, x sub 2, and x sub 3.

00:31 And that equals 5 -8.

00:36 Okay.

00:36 Now, we're going to show that these two are equivalent, the linear, the system of linear equations, and the matrix equation.

00:44 Before i multiply the matrix equation out, as we can see that these really are equivalent, let's take a look at where these numbers come from.

00:51 If you look at this first 2x3 matrix, these are the coefficients of our x -1, x -s -2, and x -s -3 from our 2, equations.

01:03 2, 3, 1 come from the equation at the top.

01:07 1 negative 4, 5 come from the second equation.

01:11 Next, we're multiplying that by a 3 by 1 matrix.

01:14 And that is where all of our variables are, x sub 1, x sub 2, and x sub 3.

01:19 And then it's going to be equal to, well, 5 and 8 are the answers to our first two equations.

01:26 So let's multiply these out.

01:29 I'm just going to remove that red that i just put on.

01:34 Okay, we have a 2x3 matrix and a 3 by 1 matrix...

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