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)

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step-by-step explanation:

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step-by-step explanation: