Solve {l}{-3x+4y+3z=16}{5x+5y+2z=13}{-4x+3y+4z=12}

$$\left\{ \begin{array}{l}{ - 3 x + 4 y + 3 z = 16 }\\{ 5 x + 5 y + 2 z = 13 }\\{ - 4 x + 3 y + 4 z = 12 }\end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=4$$

$$z=-1$$

Short Steps Using Substitution

Solve -3x+4y+3z=16 for x.

$$x=\frac{4}{3}y+z-\frac{16}{3}$$

Substitute \frac{4}{3}y+z-\frac{16}{3} for x in the second and third equation.

$$5\left(\frac{4}{3}y+z-\frac{16}{3}\right)+5y+2z=13 -4\left(\frac{4}{3}y+z-\frac{16}{3}\right)+3y+4z=12$$

Solve these equations for z and y respectively.

$$z=\frac{17}{3}-\frac{5}{3}y y=4$$

Substitute 4 for y in the equation z=\frac{17}{3}-\frac{5}{3}y.

$$z=\frac{17}{3}-\frac{5}{3}\times 4$$

Calculate z from z=\frac{17}{3}-\frac{5}{3}\times 4.

$$z=-1$$

Substitute -1 for z and 4 for y in the equation x=\frac{4}{3}y+z-\frac{16}{3}.

$$x=\frac{4}{3}\times 4-1-\frac{16}{3}$$

Calculate x from x=\frac{4}{3}\times 4-1-\frac{16}{3}.

$$x=-1$$

The system is now solved.

$$x=-1 y=4 z=-1$$