What is the solution of the system of equations?

⎧ 3x+2y+z=7

⎪

⎨ 5x+5y+4z=3

⎪

⎩ 3x+2y+3z=1

The solution of the system is (-2, - 1, - 3)

Step-by-step explanation:

* In this system of equations we have three variables

* Equation (1) ⇒ 3x + 2y + z = 7

* Equation (2) ⇒ 5x + 5y + 4z = 3

* Equation (3) ⇒ 3x + 2y + 3z = 1

- Lets use the elimination method to solve

- Equation (1) and (equation (3) have same coefficients of x and y,

then we can start by subtracting equation (1) from equation (3) to

eliminate x and y and find z

∵ 3x + 2y + z = 7 ⇒ (1)

∵ 3x + 2y + 3z = 1 ⇒ (3)

- Subtract (1) from (3)

∴ (3x - 3x) + (2y - 2y) + (3z - z) = (1 - 7)

∴ 2z = - 6

- Divide both sides by 2

∴ z = - 3

- Substitute the value of z in equations (1) and (2)

∵ 3x + 2y + (-3) = 1

∴ 3x + 2y - 3 = 1

- Add 3 for both sides

∴ 3x + 2y = 4 ⇒ (4)

∵ 5x + 5y + 4 (-3) = 3

∴ 5x + 5y - 12 = 3

- Add 12 to both sides

∴ 5x + 5y = 15 ⇒ (5)

- Now we have system of equations of two variables

∵ 3x + 2y = 4 ⇒ (4)

∵ 5x + 5y = 15 ⇒ (5)

- Multiply equation (4) by - 5 and equation (5) by 2 to eliminate y

∵ - 5 (3x) + - 5 (2y) = - 5 (4)

∵ 2 (5x) + 2 (5y) = 2 (15)

∵ - 15x - 10y = - 20 ⇒ (6)

∵ 10x + 10y = 30 ⇒ (7)

- Add equations (6) and (7)

∴ - 5x = 10

- Divide both sides by - 5

∴ x = - 2

- Substitute the value of x in equation (4) or (5) to find y

∵ 3 (2) + 2y = 4

∴ 6 + 2y = 4

- Subtract 6 from both sides

∴ 2y = - 2

- Divide both sides by 2

∴ y = - 1

* The solution of the system is (-2, - 1, - 3)