solve the following system of linear equations in three variables 3x-2y+z=2,2x+3y-z=5,x+y+z=6.
Answers
Answer:
Hey !
Answer :
x = 1 ; y = 2 ; z = 3
\begin{gathered}\\\end{gathered}
Solution:
Given 3 equations with 3 unknown variables, which is enough to find them.
3x-2y+z = 2. ------[1]
2x+3y-z= 5. -------[2]
x+y+z=6 ------------[3]
\begin{gathered}\\\end{gathered}
Step-1:
Get two equations by eliminating a variable,
(Eliminating y)
Do 2x[3]
2x+2y+2z = 12 ------[4]
Do [4]+[1]
=> 5x+3z = 14 -------[5]
Do 3×[3]
=> 3x+3y+3z = 18 -----[6]
Do [6]-[2]
=> x + 4z = 13 -----[7]
Y is eliminated in both [5] and [7]
\begin{gathered}\\\end{gathered}
Step-2: Get value of one variable using the obtained 2 equations after eliminating another variable.
(Eliminating z)
Do 5×[7]
=> 5x+20z = 65 -----[8]
Do [8]-[5]
=> 17 z = 51
=> z = 3 .
\begin{gathered}\\\end{gathered}
Step-3:
Use the equation to obtain another variable
Substitute in [5]
=> 5x +3(3) = 14
=> 5x = 14 - 9
=> 5x =5
=> x = 1.
\begin{gathered}\\\end{gathered}
Step-4:
Use another equation to obtain value of variable left.
Substitute x and z in [3]
=> 1+y+3 = 6
=> y = 6-4
=> y = 2
•°• x = 1 ; y = 2 ; z= 3
Answer:
We will check each equation by substituting in the values of the ordered triple for \displaystyle x,yx,y, and \displaystyle zz.
x
+
y
+
z
=
2
(
3
)
+
(
−
2
)
+
(
1
)
=
2
True
6
x
−
4
y
+
5
z
=
31
6
(
3
)
−
4
(
−
2
)
+
5
(
1
)
=
31
18
+
8
+
5
=
31
True
5
x
+
2
y
+
2
z
=
13
5
(
3
)
+
2
(
−
2
)
+
2
(
1
)
=
13
15
−
4
+
2
=
13
True
The ordered triple \displaystyle \left(3,-2,1\right)(3,−2,1) is indeed a solution to the system. you can do like this only but not same