Solve the following system of equations: 2x - y + z = - 3, 2x + 2y + 3z = 2, 3x - 3y - z = - 4.

Solution:

We have a system of linear equations of three variables.

Given:

2x - y + z = - 3 ---------> (1)

⇒ 2x + 2y + 3z = 2 ---------> (2)

⇒ 3x - 3y - z = - 4 ---------> (3)

Let us solve them using the substitution method.

By solving equation [3] for the variable z, we get

⇒ z = 3x - 3y + 4 …….be eq (4)

Substitute the value of z = 3x - 3y + 4 in equation [1]

⇒ 2x - y + (3x - 3y + 4) = - 3

⇒ 5x - 4y = -7 ………(multiply the equation by 11)

⇒ 55x - 44y = - 77 ---------> (5)

Substitute the value of z = 3x - 3y + 4 in equation [2]

⇒ 2x + 2y + 3 (3x - 3y + 4) = 2

⇒ 11x - 7y = - 10…….. (multiply the equation by 5)

⇒ 55x - 35y = - 50--------->  (6)

We will use elimination method and subtract eq (5) from (6)

⇒ (55x - 35y = - 50) - (55x - 44y = - 77)

⇒ 9y = 27 or y = 3

Put the value of y = 3 in equation (5) to get the value of x.

⇒ 55x - 44(3) = - 77

⇒ 55x = - 77 + 132

⇒ x = 1

Put the values of x and y in equation (4)

z = 3 (1) - 3 (3) + 4

z = 3 - 9 + 4

z = -2

Thus on solving the equations we get x = 1, y = 3 and z = -2

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Solve the following system of equations: 2x - y + z = - 3, 2x + 2y + 3z = 2, 3x - 3y - z = - 4.

Summary:

By solving the system of linear equations 2x - y + z = - 3, 2x + 2y + 3z = 2, 3x - 3y - z = - 4; we get (x, y, z) = (1, 3, -2)