Solve {l}{a+b+c=1}{a-c=-2}{-a+2b+c=4}

$$\left. \begin{array} { l } { a + b + c = 1 } \\ { a - c = - 2 } \\ { - a + 2 b + c = 4 } \end{array} \right.$$

Solve for a, b, c

$$a=-1$$

$$b=1$$

$$c=1$$

Short Steps Using Substitution

Solve a+b+c=1 for a.

$$a=-b-c+1$$

Substitute -b-c+1 for a in the second and third equation.

$$-b-c+1-c=-2 -\left(-b-c+1\right)+2b+c=4$$

Solve these equations for b and c respectively.

$$b=-2c+3 c=-\frac{3}{2}b+\frac{5}{2}$$

Substitute -2c+3 for b in the equation c=-\frac{3}{2}b+\frac{5}{2}.

$$c=-\frac{3}{2}\left(-2c+3\right)+\frac{5}{2}$$

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

$$c=1$$

Substitute 1 for c in the equation b=-2c+3.

$$b=-2+3$$

Calculate b from b=-2+3.

$$b=1$$

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

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

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

$$a=-1$$

The system is now solved.

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