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

##### 1 Answer

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

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-5x - 15y + 10z = -5

5x + 16y - 5z = -5

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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

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5x + 16y -5z = -5

-5x -10y - 95z = 205

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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

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-6y -30z = 60

6y - 100z = 200

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0 - 130z = 260, divide both sides by -130 to isolate z

-130 -130

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**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

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**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