Solve each system with dependent equations.?

Question

x+y+z=4 
-2x-y+2z=6 
x+2y+5z=18

Answer 1

The system of equations are .

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

To get two equations 1 and 2 that contain opposite coefficient of x - variable multiply the first equation by 2.

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

The resultant equation is taken as fourth equation : y + 4z = 14.

To get two equations 2 and 3 that contain opposite coefficient of x - variable multiply the third equation by 2.

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 : 3y + 12z = 42.

Solve the system of two equations with two variables.

To get two equations 4 and 5 that contain opposite coefficient of y - variable multiply the fourth equation by negative 3.

Write the equations 4 and 5 in column form and add the corresponding columns to eliminate y - variable.

.

The resultant statement 0 = 0 is false, so the system of equation has infinitely many solutions.

Let z = t, where t is any real number.

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

The fourth equation: y + 4z = 14.

y + 4(t) = 14

y = - 4t + 14.

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

The first equation: x + y + z = 4.

x + (- 4t + 14) + (t) = 4

x - 3t + 14 = 4

x - 3t = - 10

x = 3t - 10.

The solution (x, y, z) = (3t - 10, - 4t + 14, t), where t is any real number.