Write System of Equations 2

[harpazo]

The reduced row echelon form of a system of linear equations is given. Write a system of equations corresponding to the given matrix. Determine whether the system is consistent or inconsistent. If it is consistent find the solution.

Solution:

x + 0y + 4z = 4
0x + y + 3z = 2
0x + 0y + 0z = 0

Correct?

 

[HallsofIvy]

So x+ 4z= 4, y+ 3z= 2, and 0z= 0. That last equation is true for any value of z. But then we can write x= 4- 4z and y= 2- 3z. How would you write the general solution?

 

[harpazo]

So x+ 4z= 4, y+ 3z= 2, and 0z= 0. That last equation is true for any value of z. But then we can write x= 4- 4z and y= 2- 3z. How would you write the general solution?

General Solution:

x + 4z = 4
y + 3z = 2

 

[Otis]

General Solution:
x + 4z = 4
y + 3z = 2

That's not a general solution. A general solution is a statement that lists each variable separately, along with its value (or expression). Halls already told you that z can be any Real number (therefore, the system has infinite solutions).

x = ?
y = ?
z = any Real number

Symbol z is the parameter (sometimes called a 'free variable' because it can take on any value). Symbols x and y will each be defined in terms of z.

To complete the general solution, solve your two equations above (the first one for x and the second one for y).

You may want to consider reviewing general solutions and parameterizations from intermediate algebra (i.e., from solving systems using the substitution or elimination method). That way, you'll understand what books or instructors are talking about, in these beginning topics of linear algebra.

I'm not sure what your source is for this exercise, but you can google to find linear algebra videos for studying the topic from the beginning.

?

 

[harpazo]

That's not a general solution. A general solution is a statement that lists each variable separately, along with its value (or expression). Halls already told you that z can be any Real number (therefore, the system has infinite solutions).

x = ?
y = ?
z = any Real number

Symbol z is the parameter (sometimes called a 'free variable' because it can take on any value). Symbols x and y will each be defined in terms of z.

To complete the general solution, solve your two equations above (the first one for x and the second one for y).

You may want to consider reviewing general solutions and parameterizations from intermediate algebra (i.e., from solving systems using the substitution or elimination method). That way, you'll understand what books or instructors are talking about, in these beginning topics of linear algebra.

I'm not sure what your source is for this exercise, but you can google to find linear algebra videos for studying the topic from the beginning.

?

General Solution:

x = 4 - 4z
y = 2 - 3z
z = R, where R is any real number

 

[HallsofIvy]

Equivalently, x= 4- 4t, y= 2- 3t, z= t where t can be any real number. (Since "R" is customarily used to mean the set of all real numbers, I wouldn't use it to represent a specific real number.)

 

[Otis]

Harpazo, I'm sure that systems of equations and parameterized forms of solutions have been covered before Chapter 8. Are you jumping back and forth in the text, again?

 

[Dr.Peterson]

General Solution:

x = 4 - 4z
y = 2 - 3z
z = R, where R is any real number

I checked what Sullivan's style is in the Precalculus book (section 10.2), and they use this form:

[MATH]\left\{\begin{matrix}x = 4 - 4z\\ y = 2 - 3z\end{matrix}\right. \text{where }z\text{ can be any real number}[/MATH]​

So this is what you should be familiar with from reading your text.

 

[harpazo]

Harpazo, I'm sure that systems of equations and parameterized forms of solutions have been covered before Chapter 8. Are you jumping back and forth in the text, again?

No. Not jumping sections or chapters in my textbook and/or the Bible as well.

 

[harpazo]

I checked what Sullivan's style is in the Precalculus book (section 10.2), and they use this form:

[MATH]\left\{\begin{matrix}x = 4 - 4z\\ y = 2 - 3z\end{matrix}\right. \text{where }z\text{ can be any real number}[/MATH]​

So this is what you should be familiar with from reading your text.

My answer is correct with the exception that I should have used t in place of R.