Question

In the following exercises, solve the system of equations.

$\left\{\begin{array}{l}4x-3y+z=7\\ 2x-5y-4z=3\\ 3x-2y-2z=-7\end{array}\right.$

Short answer

The solution is $(-3,-5,4)$.

Step-by-step solution

Step 1. Given Information 

We are given system of linear equations,

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

Step 2. Eliminating the same variable 

Multiplying by $4$in first equation, we get

$$\begin{gathered} 4(4x-3y+z)=4\times 7 \\ 16x-12y+4z=28-\left(4\right) \end{gathered}$$

Now, adding the fourth equation and second equation, we get

$$\begin{gathered} 16x+2x-12y-5y+4z-4z=28+3 \\ 18x-17y=31-\left(5\right) \end{gathered}$$

Now, multiplying $2$in the first equation, we get

$$\begin{gathered} 2(4x-3y+z)=2\times 7 \\ 8x-6y+2z=14-\left(6\right) \end{gathered}$$

Adding the sixth and third equations, we get

$$\begin{gathered} 8x+3x-6y-2y+2z-2z=14-7 \\ 11x-8y=7-\left(7\right) \end{gathered}$$

Step 3. Solving the new system of equation 

Multiplying by $8$in fifth equation and $17$in the seventh equation, we get

localid="1644662841027" $\begin{array}{l}8(18x-17y)=8\left(31\right)\\ 144x-136y=248-\left(8\right)\\ \begin{array}{l}17(11x-8y)=17\left(7\right)\\ 187x-136y=119-\left(9\right)\end{array}\end{array}$

Now, subtracting the eighth and ninth equations, we get

$$\begin{gathered} 187x-144x-136y+136y=119-248 \\ 43x=-129 \\ x=\frac{-129}{43} \\ x=-3 \end{gathered}$$

Now, putting the value of $x$in the seventh equation, we get

$$\begin{gathered} 11x-8y=7 \\ 11(-3)-8y=7 \\ -33-8y=7 \\ y=\frac{-40}{8} \\ y=-5 \end{gathered}$$

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

$$\begin{gathered} 4x-3y+z=7 \\ 4(-3)-3(-5)+z=7 \\ -12+15+z=7 \\ 3+z=7 \\ z=7-3 \\ z=4 \end{gathered}$$

Step 4. Checking the solution

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

$$\begin{gathered} 4x-3y+z=7 \\ 4(-3)-3(-5)+4=7 \\ -12+15+4=7 \\ 7=7 \\ 2x-5y-4z=3 \\ 2(-3)-5(-5)-4\left(4\right)=3 \\ -6+25-16=3 \\ 3=3 \\ 3x-2y-2z=-7 \\ 3(-3)-2(-5)-2\left(4\right)=-7 \\ -9+10-8=-7 \\ -7=-7 \end{gathered}$$

Hence the solution is correct.