Finding the value of a variable in a matrix

[incxx]

Homework Statement

Given :

 x+y+5z = 2

 x+2y+7z = 1

 2x−y+4z = a
.
a) Determine the value of a which will make the given system have many solutions. Explain your answer.
b) Choose a value of a which will make the given system have NO solutions. Explain your answer.
c) Is it possible to choose a value of a, which will make the given system have exactly one solutions? Explain
your answer.

Homework Equations

The Attempt at a Solution

I found the reduced matrix of the system of equations and got,
1 0 3
0 1 2
0 0 0
, but am not sure on what to do next.

 

[Mark44]

Homework Statement

Given :

 x+y+5z = 2

 x+2y+7z = 1

 2x−y+4z = a
.
a) Determine the value of a which will make the given system have many solutions. Explain your answer.
b) Choose a value of a which will make the given system have NO solutions. Explain your answer.
c) Is it possible to choose a value of a, which will make the given system have exactly one solutions? Explain
your answer.

Homework Equations

The Attempt at a Solution

I found the reduced matrix of the system of equations and got,
1 0 3
0 1 2
0 0 0
, but am not sure on what to do next.

You need to set up an augmented matrix whose fourth column contains the constants on the right sides of your equations.

 

[HallsofIvy]

Equivalently, don't use matrices at all!
You have:
x+y+5z = 2
 x+2y+7z = 1
 2x−y+4z = a

Subtract the first equation from the second to get y+ 2z= -1.
Subtract the third equation from twice the first equation to get 3y+ 6z= 4- a

Subtract three times the first of those equations from the second to get 0= 7- a.

For what value of a is that true? So for what value of a does the original set of equations have an infinite number or solutions? For what values of a does it have no solution?

 

[incxx]

thank you I understand how to do it now!