Solve {l}{a_{1}+a_{2}=0}{2a_{1}+T=20}{5a_{2}+T=50}

$$\left. \begin{array}{l}{ a _ { 1 } + a _ { 2 } = 0 }\\{ 2 a _ { 1 } + T = 20 }\\{ 5 a _ { 2 } + T = 50 }\end{array} \right.$$

Solve for a_1, a_2, T

$$a_{1} = -\frac{30}{7} = -4\frac{2}{7} \approx -4.285714286$$

$$a_{2} = \frac{30}{7} = 4\frac{2}{7} \approx 4.285714286$$

$$T = \frac{200}{7} = 28\frac{4}{7} \approx 28.571428571$$

Short Steps Using Substitution

Solve a_{1}+a_{2}=0 for a_{1}.

$$a_{1}=-a_{2}$$

Substitute -a_{2} for a_{1} in the equation 2a_{1}+T=20.

$$2\left(-1\right)a_{2}+T=20$$

Solve the second equation for a_{2} and the third equation for T.

$$a_{2}=-10+\frac{1}{2}T T=50-5a_{2}$$

Substitute -10+\frac{1}{2}T for a_{2} in the equation T=50-5a_{2}.

$$T=50-5\left(-10+\frac{1}{2}T\right)$$

Solve T=50-5\left(-10+\frac{1}{2}T\right) for T.

$$T=\frac{200}{7}$$

Substitute \frac{200}{7} for T in the equation a_{2}=-10+\frac{1}{2}T.

$$a_{2}=-10+\frac{1}{2}\times \frac{200}{7}$$

Calculate a_{2} from a_{2}=-10+\frac{1}{2}\times \frac{200}{7}.

$$a_{2}=\frac{30}{7}$$

Substitute \frac{30}{7} for a_{2} in the equation a_{1}=-a_{2}.

$$a_{1}=-\frac{30}{7}$$

The system is now solved.

$$a_{1}=-\frac{30}{7} a_{2}=\frac{30}{7} T=\frac{200}{7}$$