Chapter 3.4, Problem 2CYU

To determine

To solve: the given system of equations

Answer to Problem 2CYU

The solution of the system is, $x=4$ , $y=-6$ , $z=1$

Explanation of Solution

Given Information:

The system of equations

  $\begin{array}{l}3y-5z=-23\\ 4x+2y+3z=7\\ -2x-y-z=-3\end{array}$

Formula Used

Here, three equations with three unknown variables are given. The approach used is variable elimination and simplification.

(1) First two equations are multiplied with a factor such that one of the variables has identical coefficients.

(2) Add the modified equations to eliminate the particular variable.

(3) Then, the same process is repeated between any other combinations of two equations.

(4) This gives two equations with two unknowns which can be further solved by elimination.

(3) Substitute the obtained variable in one of the equations to get the second variable and then the third variable.

Calculation Here, the equations are,

  $3y-5z=-23$........(1)

  $4x+2y+3z=7$........(2)

  $-2x-y-z=-3$........(3)

Considering the equations (2) and (3),Multiplying each of the terms in equation (3) by2, it becomes

  $-4x-2y-2z=-6$........(4)

Now, adding corresponding terms of (4) and (2) gives,

  $\{\begin{array}{l}4x+2y+3z=7\\ -4x-2y-2z=-6\end{array}$

  $\begin{array}{l}0x+0y+3z-2z=1\\ z=1\end{array}$

Now, substituting the value of z in (1), the value of y can be obtained as,

  $\begin{array}{l}3y-5\times 1=-23\\ 3y=-18\\ y=-6\end{array}$

The value of x can be calculated by substituting the values of y and z in (1) as,

  $\begin{array}{l}-2x-\left(-6\right)-1=-3\\ -2x+5=-3\\ -2x=-8\\ x=4\end{array}$

The solution of the system is, $x=4$ , $y=-6$ , $z=1$