Solve the following system equations; 2x + 4y - 3z = - 7, 3x + y + 4z = - 12, x + 3y + 4z = 4

Solution:

We have a system of linear equations of three variables.

Given:

2x + 4y - 3z = - 7 ---------> (1)

⇒ 3x + y + 4z = - 12 ---------> (2)

⇒ x + 3y + 4z = 4 --------->(3)

Let us solve them using the substitution method.

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

⇒x = -3y - 4z + 4--------->(4)

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

⇒ 2 × (-3y - 4z + 4) + 4y - 3z = -7

⇒ -6y -8z +8 + 4y -3z =-7

⇒ - 2y - 11z = -15 --------->(5)

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

⇒ 3 × (- 3y - 4z + 4) + y + 4z = -12

⇒ -9y -12z +12 +y + 4z = -12

⇒ - 8y - 8z = -24 --------->(6)

By solving equation [2] for the value of z.

⇒ 8z = - 8y + 24

⇒ z = - y + 3---------> (7)

Substitute the value of z = - y + 3 in equation [5]

⇒ -2y - 11 × (- y + 3 ) = - 15

⇒ -2y +11 y -33 = -15

⇒ 9y = 18

Solve the equation for the value of y.

⇒ y = 2

Put the value of y = 2 in equation (7) to get the value of z.

⇒ z = - 2 + 3

⇒ z = 1

Put the value of y and z in equation (4) to find x.

⇒ x = -3 (2) - 4(1) + 4

⇒ x = -6 - 4 + 4

⇒ x = - 6

Thus the solution for the given system of equations is x = -6, y =2 and z = 1

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Solve the following system equations; 2x + 4y - 3z = - 7, 3x + y + 4z = - 12, x + 3y + 4z = 4

Summary:

By solving the system of linear equations 2x + 4y - 3z = - 7, 3x + y + 4z = - 12, x + 3y + 4z = 4; we get (x, y, z) = (- 6, 2, 1)