Question

What is the solution of the following linear system?: #x + 3y − 2z = 1 , 5x + 16y − 5z = −5 , x + 7y + 19z= −41 #?

Answer 1

Equations with 3 unknown variables.
The value of x = -3, y = 0, z = -2

Explanation:

The equations are:

x + 3y - 2z = 1 eq. 1
5x + 16y -5z = -5 eq. 2
x + 2y + 19z = -41 eq. 3

Solve the equations simultaneously

with eq. 1 and 2:

1) x + 3y - 2z = 1 , multiply this equation by -5
2) 5x + 16y -5z = -5
--------------------------
-5x - 15y + 10z = -5
5x + 16y - 5z = -5
--------------------------
0 y + 5z = -10 eq. 4

with eq. 2 and 3:

2) 5x +16y - 5z = -5
3) x + 2y + 19z = -41, multiply this equation by -5
------------------------------
5x + 16y -5z = -5
-5x -10y - 95z = 205
------------------------------
0 6y - 100z = 200 eq. 5

Then, with eq. 4 and 5

4) y + 5z = -10 , multiply this equation by -6
5) 6y -100z = 200
------------------------
-6y -30z = 60
6y - 100z = 200
------------------------
0 - 130z = 260, divide both sides by -130 to isolate z
-130 -130
--------- --------
z = -2

Finding the value of y using eq. 4

4) y + 5z = -10, substitute the value of z= -2
y + 5 (-2) = -10
y - 10 = - 10, subtract both sides by 10 to isolate y
10 10
-------- ------
y = 0

Finding the value of x using eq. 1

1) x + 3y - 2z = 1, substitute values of z = -2, and y = 0
x + 3 (0) - 2(-2) = 1 , simplify
x + 0 + 4 = 1, combine like terms
x = 1 - 4 , transposing no. changed the sign of the number
x = - 3

Checking the answers:
x = -3, y = 0, z = -2

1) x + 3y - 2z = 1
-3 + 3(0) - 2(-2) = 1
-3 + 0 + 4 = 1
-3 + 4 = 1
1 = 1

2) 5x + 16y - 5z = -5
5(-3) + 16(0) - 5(-2) = -5
-15 + 0 + 10 = -5
-15 + 10 = -5
-5 = -5

3) x + 2y + 19z = -41
-3 + 2(0) + 19(-2) = -41
-3 + 0 - 38 = -41
-41 = -41