Solve {l}{c_{1}+5c_{2}+3c_{3}=0}{-2c_{1}+6c_{2}+2c_{3}=0}{3c_{1}-c_{2}+c_{2}=0}

$$\left. \begin{array}{l}{ c _ { 1 } + 5 c _ { 2 } + 3 c _ { 3 } = 0 }\\{ - 2 c _ { 1 } + 6 c _ { 2 } + 2 c _ { 3 } = 0 }\\{ 3 c _ { 1 } - c _ { 2 } + c _ { 2 } = 0 }\end{array} \right.$$

Solve for c_1, c_2, c_3

$$c_{1}=0$$

$$c_{2}=0$$

$$c_{3}=0$$

Short Steps Using Substitution

Solve c_{1}+5c_{2}+3c_{3}=0 for c_{1}.

$$c_{1}=-5c_{2}-3c_{3}$$

Substitute -5c_{2}-3c_{3} for c_{1} in the second and third equation.

$$-2\left(-5c_{2}-3c_{3}\right)+6c_{2}+2c_{3}=0 3\left(-5c_{2}-3c_{3}\right)-c_{2}+c_{2}=0$$

Solve these equations for c_{2} and c_{3} respectively.

$$c_{2}=-\frac{1}{2}c_{3} c_{3}=-\frac{5}{3}c_{2}$$

Substitute -\frac{1}{2}c_{3} for c_{2} in the equation c_{3}=-\frac{5}{3}c_{2}.

$$c_{3}=-\frac{5}{3}\left(-\frac{1}{2}\right)c_{3}$$

Solve c_{3}=-\frac{5}{3}\left(-\frac{1}{2}\right)c_{3} for c_{3}.

$$c_{3}=0$$

Substitute 0 for c_{3} in the equation c_{2}=-\frac{1}{2}c_{3}.

$$c_{2}=-\frac{1}{2}\times 0$$

Calculate c_{2} from c_{2}=-\frac{1}{2}\times 0.

$$c_{2}=0$$

Substitute 0 for c_{2} and 0 for c_{3} in the equation c_{1}=-5c_{2}-3c_{3}.

$$c_{1}=-5\times 0-3\times 0$$

Calculate c_{1} from c_{1}=-5\times 0-3\times 0.

$$c_{1}=0$$

The system is now solved.

$$c_{1}=0 c_{2}=0 c_{3}=0$$