Question

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}3x+2y-z=2\\ 2x+y+6z=-7\\ 2x+2y-14z=17\end{array}\right.$

Short answer

The given equations$\left\{\begin{array}{l}3x+2y-z=2\\ 2x+y+6z=-7\\ 2x+2y-14z=17\end{array}\right.$ has no solution.Hence the system is inconsistent

Step-by-step solution

Step 1. Given information

The given equations$\left\{\begin{array}{l}3x+2y-z=2\\ 2x+y+6z=-7\\ 2x+2y-14z=17\end{array}\right.$

Step 2. Finding the values of x,y and z

Consider the equations $3x+2y-z=2,2x+y+6z=-7$and $2x+2y-14z=17$.To solve these equations,it seems easiest to eliminate the variable y first.So,we multiply the second equation bt 2 and the substract it from first equation

$-3x+2y-1z=24x+2y+12z=-14-x+0y-13z=16$

$-x+0y-13z=16\cdots \cdots \cdots \left(1\right)$

Again we multiplay the second equation by 2 and the substract last equation from it

$-4x+2y+12z=-142x+2y-14z=172x+0y+26z=-14$

$2x+0y+26z=-14\cdots \cdots \cdots \left(2\right)$

Now we multiplay equation (1) by -2 and then substract equation (2)

$-2x+26y=322x+26y=-140x+0y=46$

$0x+0y=46\cdots \cdots \cdots \cdots \left(3\right)$

Equation (3) has no solution.Hence the system is inconsistent