Solve the following system of equations; x + 3y - z = 2, x - 2y + 3z = 7, x + 2y - 5z = -21

Solution:

We have a system of linear equations of three variables.

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

⇒ x - 2y + 3z = 7 --------->(2)

⇒ x + 2y - 5z = -21 --------->(3)

Let us solve them using the substitution method.

By solving equation [1] for the variable z, we get

⇒ z = x + 3y - 2--------->(4)

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

⇒ x - 2y + 3(x + 3y - 2) = 7

= x - 2y + 3x + 9y - 6 = 7

⇒ 4x + 7y = 13--------->(5)

Substitute the value of z = 3x + 6y - 1 in equation [3]

⇒ x + 2y - 5(x + 3y - 2) = - 21

⇒ x + 2y -5x -15 y + 10 = -21

⇒ 4x + 13y = 31--------->(6)

Subtracting eq (5) from (6), we get

⇒ (4x + 7y = 13) - (4x + 13y = 31)

⇒ 6y = 18 or y = 3

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

⇒ 4x + 7 (3) = 13

⇒ 4x = 13 - 21

⇒ x = - 2

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

z = -2 + 3(3) -2

⇒ z = - 2 + 9 - 2

⇒ z = 5

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Solve the following system of equations; x + 3y - z = 2, x - 2y + 3z = 7, x + 2y - 5z = -21

Summary:

By solving the system of linear equations x + 3y - z = 2, x - 2y + 3z = 7, x + 2y - 5z = -21; we get (x, y, z) = (-2, 3, 5)