Question

Solve the system of equations

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

Short answer

The solution for the system of equations is,

$$\begin{gathered} x=-3 \\ y=2 \\ z=3 \end{gathered}$$

Step-by-step solution

Step 1. Given the information

The system of equations is,

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

Step 2. Eliminating z from equations (2) and (3).

Eliminating $z$from equations (2) and (3).

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

Solving the two equations we get,

role="math" localid="1644557042156" $4x+9y=6.........\left(4\right)$

Step 3. Finding the value of y by solving equations (1) and (4).

Solving equations (1) and (4).

$$\begin{gathered} 4x+10y=8........\left(1\right)\times 2 \\ 4x+9y=6.........\left(4\right) \end{gathered}$$

Subtracting the two equations we get,

$y=2$

Step 4. Finding the value of x.

Substituting $y=2$in the equation

$2x+5y=4$.

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

Step 5. Finding the value of z

Substituting $x=-3$in the equation

$4x+3z=-3$

$$\begin{gathered} 4(-3)+3z=-3 \\ -12+3z=-3 \\ 3z=-3+12 \\ 3z=9 \\ z=3 \end{gathered}$$

Step 6. Checking the solution for the equation 2x+5y=4.

Substituting $$\begin{gathered} x=-3 \\ y=2 \end{gathered}$$in the equation

$2x+5y=4$

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

This is true.

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

Substituting $$\begin{gathered} y=2 \\ z=3 \end{gathered}$$in the equation

$3y-z=3$.

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

This is true.

Step 8.Checking the solution for the equation 4x+3z=-3

Substituting $$\begin{gathered} x=-3 \\ z=3 \end{gathered}$$in the equation

$4x+3z=-3$

role="math" localid="1644558051186" $$\begin{gathered} 4(-3)+3\left(3\right)=-3 \\ -12+9=-3 \\ -3=-3 \end{gathered}$$

This is true.