Solve {l}{1/2a+v_{0}+s_{0}=132}{2a+2v_{0}+s_{0}=100}{9/2a+3v_{0}+s_{0}=36}

$$\left. \begin{array} { l } { \frac { 1 } { 2 } a + v _ { 0 } + s _ { 0 } = 132 } \\ { 2 a + 2 v _ { 0 } + s _ { 0 } = 100 } \\ { \frac { 9 } { 2 } a + 3 v _ { 0 } + s _ { 0 } = 36 } \end{array} \right.$$

Solve for a, v_0, s_0

$$a=-32$$

$$v_{0}=16$$

$$s_{0}=132$$

Short Steps Using Substitution

Solve \frac{1}{2}a+v_{0}+s_{0}=132 for v_{0}.

$$v_{0}=-\frac{1}{2}a-s_{0}+132$$

Substitute -\frac{1}{2}a-s_{0}+132 for v_{0} in the second and third equation.

$$2a+2\left(-\frac{1}{2}a-s_{0}+132\right)+s_{0}=100 \frac{9}{2}a+3\left(-\frac{1}{2}a-s_{0}+132\right)+s_{0}=36$$

Solve these equations for a and s_{0} respectively.

$$a=-164+s_{0} s_{0}=180+\frac{3}{2}a$$

Substitute -164+s_{0} for a in the equation s_{0}=180+\frac{3}{2}a.

$$s_{0}=180+\frac{3}{2}\left(-164+s_{0}\right)$$

Solve s_{0}=180+\frac{3}{2}\left(-164+s_{0}\right) for s_{0}.

$$s_{0}=132$$

Substitute 132 for s_{0} in the equation a=-164+s_{0}.

$$a=-164+132$$

Calculate a from a=-164+132.

$$a=-32$$

Substitute -32 for a and 132 for s_{0} in the equation v_{0}=-\frac{1}{2}a-s_{0}+132.

$$v_{0}=-\frac{1}{2}\left(-32\right)-132+132$$

Calculate v_{0} from v_{0}=-\frac{1}{2}\left(-32\right)-132+132.

$$v_{0}=16$$

The system is now solved.

$$a=-32 v_{0}=16 s_{0}=132$$