Solve {l}{a+b+c=2}{2a+c=-1}{-b+c=-1}

$$\left. \begin{array}{l}{ a + b + c = 2 }\\{ 2 a + c = - 1 }\\{ - b + c = - 1 }\end{array} \right.$$

Solve for a, b, c

$$a=-1$$

$$b=2$$

$$c=1$$

Short Steps Using Substitution

Solve a+b+c=2 for a.

$$a=-b-c+2$$

Substitute -b-c+2 for a in the equation 2a+c=-1.

$$2\left(-b-c+2\right)+c=-1$$

Solve the second equation for b and the third equation for c.

$$b=\frac{5}{2}-\frac{1}{2}c c=b-1$$

Substitute \frac{5}{2}-\frac{1}{2}c for b in the equation c=b-1.

$$c=\frac{5}{2}-\frac{1}{2}c-1$$

Solve c=\frac{5}{2}-\frac{1}{2}c-1 for c.

$$c=1$$

Substitute 1 for c in the equation b=\frac{5}{2}-\frac{1}{2}c.

$$b=\frac{5}{2}-\frac{1}{2}$$

Calculate b from b=\frac{5}{2}-\frac{1}{2}.

$$b=2$$

Substitute 2 for b and 1 for c in the equation a=-b-c+2.

$$a=-2-1+2$$

Calculate a from a=-2-1+2.

$$a=-1$$

The system is now solved.

$$a=-1 b=2 c=1$$