Solve the following system of equations;
4x + 7y - 2z = 0, 3x - 5y + 3z = 9, 3x + 6y - z = 1

Solution:

Given is a system of linear equations of 3 variables.

Let

x + 7y - 2z = 0 --- (1)

⇒ 3x - 5y + 3z = 9 --- (2)

⇒ 3x + 6y - z = 1 --- (3)

Let us use the substitution method for solving the system of linear equations simultaneously.

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

⇒ z = 3x + 6y - 1 --- (4)

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

⇒ 4x + 7y - 2 × (3x+6y-1) = 0

⇒ -2x - 5y = -2 --- (5)

Substitute the value of z = 3x + 6y - 1 in equation [2]

⇒ 3x - 5y + 3 × (3x+6y-1) = 9

⇒ 12x + 13y = 12 --- (6)

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

⇒ 13y = -12x + 12

⇒ y = -12x / 13 + 12 / 13 --- (7)

Substitute the value of y = -12x/13 + 12/13 in equation [1]

⇒ -2x - 5 × (-12x/13 + 12/13) = -2

⇒ 34x/13 = 34/13

⇒ 34x = 34

By solving equation [1] for the value of x.

⇒ 34x = 34

x = 1

Put the value of x = 1 in equation (4) and (7) to get the values of y and z.

x = 1

y = -12x/13 + 12/13

z = 3x + 6y - 1

Thus, y = -(12/13) (1) + 12/13 = 0

 z = 3(1) + 6(0) - 1 = 2

---

Solve the following system of equations;
4x + 7y - 2z = 0, 3x - 5y + 3z = 9, 3x + 6y - z = 1

Summary:

By solving the system of linear equations 4x + 7y - 2z = 0, 3x - 5y + 3z = 9, 3x + 6y - z = 1 ; we get (x, y, z) = (1, 0, 2)