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14 February, 12:31

What is the solution of the system of equations?

⎧ 3x+2y+z=7



⎨ 5x+5y+4z=3



⎩ 3x+2y+3z=1

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Answers (1)
  1. 14 February, 12:33
    0
    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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