Chapter 3, Problem 29SGR

To determine

To Solve the system of equations to find values of x , y and z .

Answer to Problem 29SGR

  $x=5;y=-5;z=-4$

Explanation of Solution

Given information: $2x-z=14$ -------- (1)

  $3x-y+5z=0$ ------ (2)

  $4x+2y+3z=-2$ ----- (3)

Method used: Use substitution method to solve the system of equations.

Calculation:

Take the first equation $2x-z=14$ and solve it for z .

Subtract both sides $2x$

  $-z=-2x+14$

Divide both sides by -1.

  $z=2x-14$

Substitute z as $2x-14$ into other two equations to get rid the term ‘ z’ .

  $\begin{array}{l}3x-y+5(2x-14)=0\\ 4x+2y+3(2x-14)=-2\end{array}$

Simplify them by distributing and combining like terms.

  $3x-y+5(2x-14)=0$

  $3x-y+10x-70=0$

  $13x-y=70$ ------ (4)

For the $4x+2y+3(2x-14)=-2$

  $4x+2y+6x-42=-2$

  $10x+2y=40$

Divide both sides by 2 to simplify the equation.

  $5x+y=20$ ------- (5)

Now, add the equations (4) and (5) to get rid the y terms.

  $\begin{array}{l}13x-y=70\\ 5x+y=20\\ --------\\ 18x=90\end{array}$

Divide both sides by 18.

  $x=5$

Substitute x as 5 into any of the equations (4) or (5).

Here, use equation (5)

  $5(5)+y=20$

  $25+y=20$

Subtract both sides 25.

  $y=-5$

Now, plug in x as 5 into equation (1) to get value for x .

  $2x-z=14$

  $2(5)-z=14$

  $10-z=14$

Subtract both sides 10.

  $-z=4$

Divide both sides by -1.

  $z=-4$

Hence $x=5;y=-5;z=-4$