Question

Solve the system of equations:

$\left\{\begin{array}{l}x+2z=0\\ 4y+3z=-2\\ 2x-5y=3\end{array}\right.$

Short answer

The solution for the system of equations is,

$$\begin{gathered} x=4 \\ y=1 \\ z=-2 \end{gathered}$$.

Step-by-step solution

Step 1. Given the information

The system of equations is,

$\left\{\begin{array}{l}x+2z=0.............\left(1\right)\\ 4y+3z=-2........\left(2\right)\\ 2x-5y=3...........\left(3\right)\end{array}\right.$

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

Eliminating $z$with the equations (1) and (2)

$$\begin{gathered} -3x-6z=0.......\left(1\right)\times -3 \\ 8y+6z=-4......\left(2\right)\times 2 \end{gathered}$$

Solving the equations we get,

$-3x+8y=4.....\left(4\right)$

Step 3. Solving the equations (3) and (4) to find the value of y.

Solving the equations (3) and (4),

$$\begin{gathered} -6x+16y=-8........\left(4\right)\times 2 \\ 6x-15y=9..........\left(3\right)\times 3 \end{gathered}$$

Adding the equations we get,

$$\begin{gathered} 16y-15y=-8+9 \\ y=1 \end{gathered}$$

Step 4. Finding the value of x.

Substituting $y=1$in the equation

$2x-5y=3$,

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

Step 5. Finding the value of z.

Substituting $x=4$in the equation

$x+2z=0$,

$$\begin{gathered} 4+2z=0 \\ 2z=-4 \\ z=-2 \end{gathered}$$

Step 6. Checking the solution for the equation x+2z=0.

Substituting $$\begin{gathered} x=4 \\ y=1 \\ z=-2 \end{gathered}$$in the equation

$x+2z=0$,

$$\begin{gathered} 4+2(-2)=0 \\ 4-4=0 \\ 0=0 \end{gathered}$$

This is true.

Step 7. Checking the solution for the equation 4y+3z=-2.

Substituting $$\begin{gathered} x=4 \\ y=1 \\ z=-2 \end{gathered}$$in the equation

$4y+3z=-2$,

$$\begin{gathered} 4\left(1\right)+3(-2)=-2 \\ 4-6=-2 \\ -2=-2 \end{gathered}$$

This is true.

Step 8. Checking the solution for the equation 2x-5y=3.

Substituting $$\begin{gathered} x=4 \\ y=1 \\ z=-2 \end{gathered}$$in the equation

$2x-5y=3$

$$\begin{gathered} 2\left(4\right)-5\left(1\right)=3 \\ 8-5=3 \\ 3=3 \end{gathered}$$

This is true.