Question

Solve :

$$\begin{gathered} 3x-4z=-1 \\ 2y+3z=2 \\ 2x+3y=6 \end{gathered}$$

Short answer

The solution of given system of equation is $(-3,4,-2)$.

Step-by-step solution

Step 1. Given information  

We are given system of linear equations,

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

Step 2. Solving the equations 

Multiplying first equation by $3$and second equation by $4$, we get

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

Now, adding the above equations, we get

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

Now, multiplying localid="1644572262771" $2$in sixth equation and $9$in third equation, we get

$$\begin{gathered} 18x+16y=10-\left(7\right) \\ 18x+27y=54-\left(8\right) \end{gathered}$$

Now, subtracting the above equations, we get

localid="1644572447926" $$\begin{gathered} 18x-18x+27y-16y=54-10 \\ 11y=44 \\ y=\frac{44}{11} \\ y=4 \end{gathered}$$

Putting the value of localid="1644573908088" $y$in third equation, we get

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

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

$$\begin{gathered} 2\left(4\right)+3z=2 \\ 8+3z=2 \\ 3z=2-8 \\ 3z=-6 \\ z=-2 \end{gathered}$$

Hence the solution is $(-3,4,-2)$.

Step 3. Checking the solution 

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

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

This is true, hence the solution is correct.