Question

In the following exercises, solve the system of equations.

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

Short answer

The solution is $(2,-3,-4)$.

Step-by-step solution

Step 1. Given Information

We are given system of linear equations,

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

Step 2. Eliminating the same variable 

Multiplying second equation by $4$, we get

$$\begin{gathered} 4(5x+2y+z)=4\times 0 \\ 20x+8y+4z=0-\left(4\right) \end{gathered}$$

Now, subtracting first and fourth equation, we get

$$\begin{gathered} 20x-3x+8y+5y+4z-4z=0-5 \\ 17x+13y=-5-\left(5\right) \end{gathered}$$

Now, multiplying the second equation by $2$, we get

$$\begin{gathered} 2(5x+2y+z)=2\times 0 \\ 10x+4y+2z=0-\left(6\right) \end{gathered}$$

Now, adding sixth and third equation, we get

$$\begin{gathered} 10x+2x+4y+3y+2z-2z=0+3 \\ 12x+7y=3-\left(7\right) \end{gathered}$$

Step 3. Solving the new system of equation 

Multiplying $13$in seventh equation and $7$in fifth equation, we get

role="math" localid="1644661096849" $$\begin{gathered} 13(12x+7y)=13\left(3\right) \\ 156x+91y=39-\left(8\right) \\ 7(17x+13y)=7(-5) \\ 119x+91y=-35-\left(9\right) \end{gathered}$$

Now, subtracting eighth and ninth equation, we get

$$\begin{gathered} 156x-119x+91y-91y=39-(-35) \\ 37x=74 \\ x=\frac{74}{37} \\ x=2 \end{gathered}$$

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

$$\begin{gathered} 12x+7y=3 \\ 12\left(2\right)+7y=3 \\ 24+7y=3 \\ 7y=3-24 \\ y=\frac{-21}{7} \\ y=-3 \end{gathered}$$

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

$$\begin{gathered} 5x+2y+z=0 \\ 5\left(2\right)+2(-3)+z=0 \\ 10-6+z=0 \\ z=-4 \end{gathered}$$

Step 4. Checking the solution

Putting the value of $x,y,z$in the equations, we get

$$\begin{gathered} 3x-5y+4z=5 \\ 3\left(2\right)-5(-3)+4(-4)=5 \\ 6+15-16=5 \\ 5=5 \\ 5x+2y+z=0 \\ 5\left(2\right)+2(-3)-4=0 \\ 10-6-4=0 \\ 0=0 \\ 2x+3y-2z=3 \\ 2\left(2\right)+3(-3)-2(-4)=3 \\ 4-9+8=3 \\ 3=3 \end{gathered}$$

Hence the solution is correct.