Solve {l}{3a-b+c=7}{2a+3b=-2}{a+b+c=-1}

$$\left\{ \begin{array} { l } { 3 a - b + c = 7 } \\ { 2 a + 3 b = - 2 } \\ { a + b + c = - 1 } \end{array} \right.$$

Solve for a, b, c

$$a=2$$

$$b=-2$$

$$c=-1$$

Short Steps Using Substitution

Solve 3a-b+c=7 for b.

$$b=3a+c-7$$

Substitute 3a+c-7 for b in the second and third equation.

$$2a+3\left(3a+c-7\right)=-2 a+3a+c-7+c=-1$$

Solve these equations for a and c respectively.

$$a=-\frac{3}{11}c+\frac{19}{11} c=3-2a$$

Substitute -\frac{3}{11}c+\frac{19}{11} for a in the equation c=3-2a.

$$c=3-2\left(-\frac{3}{11}c+\frac{19}{11}\right)$$

Solve c=3-2\left(-\frac{3}{11}c+\frac{19}{11}\right) for c.

$$c=-1$$

Substitute -1 for c in the equation a=-\frac{3}{11}c+\frac{19}{11}.

$$a=-\frac{3}{11}\left(-1\right)+\frac{19}{11}$$

Calculate a from a=-\frac{3}{11}\left(-1\right)+\frac{19}{11}.

$$a=2$$

Substitute 2 for a and -1 for c in the equation b=3a+c-7.

$$b=3\times 2-1-7$$

Calculate b from b=3\times 2-1-7.

$$b=-2$$

The system is now solved.

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