An accountant earns $35 for the first hour and $23 for each subsequent hour per client. The accountant charges a client $127. How many hours did they work?

tashiiexb0o5c
2022-09-13
Answered

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asked 2021-06-01

Find the linear approximation of the function

Use L(x) to approximate the numbers

asked 2022-06-19

I have a system of linear equations:

$x-y+2z-t=1$

$2x-3y-z+t=-1$

$x+(\alpha -4)z=\alpha -3$

I have already found that this system has a solution for any value of $\alpha $. Now I need to find the $\alpha $ for which the matrix of the system has a rank=2. The row echelon form of the matrix looks like this:

$\left[\begin{array}{ccccccccc}1& & -1& & 2& & -1& & 1\\ 0& & -1& & -5& & 3& & -3\\ 0& & 0& & \alpha -11& & 4& & \alpha -7\end{array}\right]$

I don't think that the rank of this matrix could be 2 for any value of $\alpha $ but the problem specifically asks for me to prove that it can. Maybe I'm missing something. Any help is appreciated.

$x-y+2z-t=1$

$2x-3y-z+t=-1$

$x+(\alpha -4)z=\alpha -3$

I have already found that this system has a solution for any value of $\alpha $. Now I need to find the $\alpha $ for which the matrix of the system has a rank=2. The row echelon form of the matrix looks like this:

$\left[\begin{array}{ccccccccc}1& & -1& & 2& & -1& & 1\\ 0& & -1& & -5& & 3& & -3\\ 0& & 0& & \alpha -11& & 4& & \alpha -7\end{array}\right]$

I don't think that the rank of this matrix could be 2 for any value of $\alpha $ but the problem specifically asks for me to prove that it can. Maybe I'm missing something. Any help is appreciated.

asked 2022-02-22

In describing the solution of a system of linear equations with many solutions, why do we use a free variable as a parameter to describe the other variables in the solution? Why do we not we use a leading variable? Since by the commutative property of addition we can swap between the free and leading variables, e.g. x + y + z = x + z + y; the solution set will essentially be the same (albeit having different orders).

Definitions:

1.A free variable is a parameter that is not a leading variable.

2.A leading variable is the first variable that has a non-zero coefficient in reduced form.

3.These definitions are most easily understood with respect to the Echelon form of a system of linear equations expressed as a Matrix.

For example:

Let S be the solution set of the system

$x+y+z=3$

$y-z=4$

Using the free variable z as the parameter

$S=\{(-2z-1,z+4,z)\mid z\in \mathbb{R}\}$ .

Using the leading variable y as the parameter

$S=\{(-2y+7,y,y-4)\mid y\in \mathbb{R}\}$ .

Definitions:

1.A free variable is a parameter that is not a leading variable.

2.A leading variable is the first variable that has a non-zero coefficient in reduced form.

3.These definitions are most easily understood with respect to the Echelon form of a system of linear equations expressed as a Matrix.

For example:

Let S be the solution set of the system

Using the free variable z as the parameter

Using the leading variable y as the parameter

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