Question

Solve:

$$\begin{gathered} 4x-3z=-5 \\ 3y+2z=7 \\ 3x+4y=6 \end{gathered}$$

Short answer

The solution of given system of equation is $(-2,3,-1)$.

Step-by-step solution

Step 1. Given information 

We are given system of linear equations

$$\begin{gathered} 4x-3z=-5-\left(1\right) \\ 3y+2z=7-\left(2\right) \\ 3x+4y=6-\left(3\right) \end{gathered}$$

Step 2. Solving equations 

Multiplying $2$in first equation and $3$in second equation, we get

$$\begin{gathered} 8x-6z=-10 \\ 9y+6z=21 \end{gathered}$$

Adding the above equations,

$$\begin{gathered} 8x+9y-6z+6z=-10+21 \\ 8x+9y=11-\left(4\right) \end{gathered}$$

Now, multiplying $8$in third equation and $3$in fourth equation, we get

$$\begin{gathered} 24x+32y=48 \\ 24x+27y=33 \end{gathered}$$

Subtracting the above equations, we get

$$\begin{gathered} 24x-24x+32y-27y=48-33 \\ 5y=15 \\ y=3 \end{gathered}$$

Putting the value of $y$in third equation, we get

$$\begin{gathered} 3x+4y=6 \\ 3x+4\left(3\right)=6 \\ 3x+12=6 \\ 3x=-6 \\ x=-2 \end{gathered}$$

Now, putting the value of $y$in second equation, we get

$$\begin{gathered} 3y+2z=7 \\ 3\left(3\right)+2z=7 \\ 9+2z=7 \\ 2z=-2 \\ z=-1 \end{gathered}$$

Hence the solution is $(-2,3-1)$.

Step 3. Checking the solution 

Checking the solution by putting the value of $x,y,z$in the equations, we get

$$\begin{gathered} 4x-3z=-5 \\ 4(-2)-3(-1)=-5 \\ -8+3=-5 \\ -5=-5 \\ 3y+2z=7 \\ 3\left(3\right)+2(-1)=7 \\ 9-2=7 \\ 7=7 \\ 3x+4y=6 \\ 3(-2)+4\left(3\right)=6 \\ -6+12=6 \\ 6=6 \end{gathered}$$

The solution satisfies all the equations, hence the solution is correct.