# 1) solve by elimination 2) solve by substitution?

1) -2x + 2y + 3z = 0
-2x + y + z = -3
2x + 3y + 3z = 5

2) -x - y - z = -8
-4x + 4y + 5z = 7
2x + 2z = 4

Elimination Method :

(1). The system of equations are .

Use the elimination method to make a system of two equations in two variables.

The two equations 1 and 2 contains same coefficient of x - variable.

Write the equations 1 and 2 in column form and subtract the corresponding columns to eliminate x - variable.

The resultant equation is taken as fourth equation : .

The two equations 2 and 3 contains opposite coefficient of x - variable.

Write the equations 2 and 3 in column form and add the corresponding columns to eliminate x - variable.

The resultant equation is taken as fifth equation : .

Solve the system of two equations with two variables.

Neither variable has a common coefficient in equation 4 and 5. The coefficient  of the z - variables are 2 and 4 and their least common multiple is 4, so multiply each equation by the value  that will make the z - coefficient 4.

To get two equations 4 and 5 that contain opposite terms multiply the fourth equation by negative 2.

Write the equations in column form and add the corresponding columns to eliminate z - variable.

.

The resultant equation is .

Use one of the equation with two variables (Equation: 4 or 5) to solve for z.

The fourth equation : .

.

Solve for x using one of the original equations with three variables.

The third equation: .

.

The solution .

Substitution Method :

(2). The system of equations are - x - y - z = - 8, - 4x + 4y + 5z = 7 and 2x + 2z = 4.

Step 1 : Solve the equation 3 : 2x + 2z = 4 x + z = 2 for x since the coefficient is 1 and this is two variable equation.

Subtract z from each side.

x + z - z = 4 - z

x = 2 - z

Step 2 : Substitute 2 - z for x in the equation 2 : - x - y - z = - 8 to find the value of y.

- (2 - z) - y - z = - 8

- 2 + z - y - z = - 8

- 2 - y = - 8

- 2 - y + 2 = - 8 + 2

- y = - 6

Multiply each side by negative 1.

y = 6.

Step 3 : Substitute 6 for y and 2 - z for x in equation 2 : - 4x + 4y + 5z = 7 to find z.

- 4(2 - z) - 4(6) + 5z = 7

Apply distributive property : a(b + c) = ab + ac.

- 8 + 4z + 24 + 5z = 7

9z + 16 = 7

Subtract 16 from each side.

9z + 16 - 16 = 7 - 16

9z = - 9

Divide each side by 9.

z = - 1.

Step 4 : Substitute - 1 for z in equation 3 : 2x + 2z = 4 x + z = 2 to find x.

x + (- 1) = 2

x - 1 + 1 = 2 + 1

x = 3.

The solution (x, y, z) = (3, 6, - 1).