Solve {c}{4a+3b+5c=-15}{a-b+2c=-7}{2a+b+3c=-9}

$$\left\{ \begin{array} { c } { 4 a + 3 b + 5 c = - 15 } \\ { a - b + 2 c = - 7 } \\ { 2 a + b + 3 c = - 9 } \end{array} \right.$$

Solve for a, b, c

$$a=-2$$

$$b=1$$

$$c=-2$$

Short Steps Using Substitution

Reorder the equations.

$$a-b+2c=-7 4a+3b+5c=-15 2a+b+3c=-9$$

Solve a-b+2c=-7 for a.

$$a=b-2c-7$$

Substitute b-2c-7 for a in the second and third equation.

$$4\left(b-2c-7\right)+3b+5c=-15 2\left(b-2c-7\right)+b+3c=-9$$

Solve these equations for b and c respectively.

$$b=\frac{3}{7}c+\frac{13}{7} c=3b-5$$

Substitute \frac{3}{7}c+\frac{13}{7} for b in the equation c=3b-5.

$$c=3\left(\frac{3}{7}c+\frac{13}{7}\right)-5$$

Solve c=3\left(\frac{3}{7}c+\frac{13}{7}\right)-5 for c.

$$c=-2$$

Substitute -2 for c in the equation b=\frac{3}{7}c+\frac{13}{7}.

$$b=\frac{3}{7}\left(-2\right)+\frac{13}{7}$$

Calculate b from b=\frac{3}{7}\left(-2\right)+\frac{13}{7}.

$$b=1$$

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

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

Calculate a from a=1-2\left(-2\right)-7.

$$a=-2$$

The system is now solved.

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