Question

Solve the system of equations.

$\left\{\begin{array}{l}3x-z=-3\\ 5y+2z=-6\\ 4x+3y=-8\end{array}\right.$

Short answer

The solution for the system of equations is,

$$\begin{gathered} x=-2 \\ y=0 \\ z=-3 \end{gathered}$$

Step-by-step solution

Step 1. Given the information.

The system of equations is,

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

Step 2. Eliminating z from equations (1) and (2).

Eliminating $z$from equations (1) and (2).

$$\begin{gathered} 6x-2z=-6........\left(1\right)\times 2 \\ 5y+2z=-6........\left(2\right) \end{gathered}$$

Solving the equations we get,

$6x+5y=-12......\left(4\right)$

Step 3. Finding the value of y.

Solving the equations (3) and (4).

$$\begin{gathered} 12x+9y=-24.......\left(3\right)\times 3 \\ 12x+10y=-24.....\left(4\right)\times 2 \end{gathered}$$

Subtracting the equations we get,

$$\begin{gathered} -y=0 \\ y=0 \end{gathered}$$

Step 4. Finding the value of x.

Substituting $y=0$in the equation

$4x+3y=-8$.

$$\begin{gathered} 4x+3\left(0\right)=-8 \\ 4x+0=-8 \\ 4x=-8 \\ x=-2 \end{gathered}$$

Step 5. Finding the value of z.

Substituting $y=0$in the equation

$5y+2z=-6$

$$\begin{gathered} 5\left(0\right)+2z=-6 \\ 0+2z=-6 \\ 2z=-6 \\ z=-3 \end{gathered}$$

Step 6. Checking the solution for the equation 3x-z=-3

Substituting $$\begin{gathered} x=-2 \\ z=-3 \end{gathered}$$in the equation

$3x-z=-3$.

$$\begin{gathered} 3(-2)-(-3)=-3 \\ -6+3=-3 \\ -3=-3 \end{gathered}$$

This is true.

Step 7. Checking the solution for the equation 5y+2z=-6.

Substituting $$\begin{gathered} y=0 \\ z=-3 \end{gathered}$$in the equation

$5y+2z=-6$

$$\begin{gathered} 5\left(0\right)+2(-3)=-6 \\ 0-6=-6 \\ -6=-6 \end{gathered}$$

This s true.

Step 8. Checking the solution for the equation 4x+3y=-8.

Substituting $$\begin{gathered} x=-2 \\ y=0 \end{gathered}$$in the equation

$4x+3y=-8$

$$\begin{gathered} 4(-2)+3\left(0\right)=-8 \\ -8+0=-8 \\ -8=-8 \end{gathered}$$

This is true.