alegra help?

Question

hi guys can you help with step by step instructions how to do these i have like 50 problems simular to these so if you show me how to do these I think I can tackle the rest.:
1.Solve the system of equations.

x + y = 4
y + z = 5
x + z = 5


2.Write the augmented matrix for the given system of equations.
x + y + z = 12
y + z = 7
x + y = 11


3.For the given augmented matrix write the system of equations. Use x,y,z as variables.

[1 -2 1 | 3]
[3 1 0| 11]
[0 0 2| 4]

Answer 1

will do one for you

1. lets substitute into x + y = 4
so what does y= ?? it's in the second equation = 5-z
(we subtracted Z from both sides - still an equation)
what does x = ---- 3rd equation = 5-z
(same deal)
x + y = 4
(5-z) + (5-z) = 4
10 - 2z = 4..add 2z
10 = 4 + 2z....subtract 4
6=2z
z=3

now you can solve the second and third equations for x and y

Answer 2

1. solve the first equation for x to get x = 4 - y
next plug this into the third equation in place of x to get
(4 - y) + z = 5
now solve this equation for z to get z = 5 - 4 + y ....or z = 1 + y
plug this into the second equation for z and solve for y
y + (1 + y) = 5
2y +1 = 5
2y = 4
y = 2
now that you know y = 2 use that to solve the first equation for x...... x = 4 - y
x = 4 - 2
x = 2
and the second equation for z......
y + z = 5
2 + z = 5
z = 3

problem 2
for an augmented matrix you must know that a matrix is made up of rows and columns. the rows run horizontally and the columns run vertically. for an augmented matrix the variables go into the left side of the matrix for x, y and z in columns 1, 2 and 3 respectively and the number on the right side of the equation goes on the right side of the matrix. so for your problem you would have an augmented matrix that looked like:
[1 1 1 | 12]
[0 1 1 | 7 ]
[1 0 1 | 11]


problem 3
work number two backwards. the first number in the first row represents the x variable, the second represents the y and the third represents the z. therefore your system of equations would be:
1x -2y +1z = 3 or simply .... x - 2y + z = 3
3x + y + 0z = 11 or just ........3x + y = 11
0x + 0y + 2z = 4 or just ........ 2z = 4

good luck with the rest of the problems!!

Answer 3

Question 1
- x - z = - 5
y + z = 5----ADD
- x + y = 0

x + y = 4
-x + y = 0------ADD
2y = 4
y = 2
x = 2
z = 3
Question 2
[ 1 1 1 ] = [12]
[ 0 1 1 ] = [ 7]
[ 1 1 0 ] = [11]
Question 3
x - 2y + z = 3
3x + y = 11
2z = 4

Answer 4

Q1
x=2
y=2
z=3

Q2
X=5
Y=6
Z=1

Q3
x=3
y=2
z=4