Solve {l}{-alpha-1/4beta=-1}{3/4beta=1-t}{2/3alpha+1/4beta=t}

$$\left\{ \begin{array} { l } { - \alpha - \frac { 1 } { 4 } \beta = - 1 } \\ { \frac { 3 } { 4 } \beta = 1 - t } \\ { \frac { 2 } { 3 } \alpha + \frac { 1 } { 4 } \beta = t } \end{array} \right.$$

Solve for α, β, t

$$t=\frac{7}{10}=0.7$$

$$\alpha =\frac{9}{10}=0.9$$

$$\beta =\frac{2}{5}=0.4$$

Short Steps Using Substitution

Reorder the equations.

$$\frac{2}{3}\alpha +\frac{1}{4}\beta =t \frac{3}{4}\beta =1-t -\alpha -\frac{1}{4}\beta =-1$$

Solve \frac{2}{3}\alpha +\frac{1}{4}\beta =t for t.

$$t=\frac{2}{3}\alpha +\frac{1}{4}\beta$$

Substitute \frac{2}{3}\alpha +\frac{1}{4}\beta for t in the equation \frac{3}{4}\beta =1-t.

$$\frac{3}{4}\beta =1-\left(\frac{2}{3}\alpha +\frac{1}{4}\beta \right)$$

Solve the second equation for \beta and the third equation for \alpha .

$$\beta =1-\frac{2}{3}\alpha \alpha =-\frac{1}{4}\beta +1$$

Substitute 1-\frac{2}{3}\alpha for \beta in the equation \alpha =-\frac{1}{4}\beta +1.

$$\alpha =-\frac{1}{4}\left(1-\frac{2}{3}\alpha \right)+1$$

Solve \alpha =-\frac{1}{4}\left(1-\frac{2}{3}\alpha \right)+1 for \alpha .

$$\alpha =\frac{9}{10}$$

Substitute \frac{9}{10} for \alpha in the equation \beta =1-\frac{2}{3}\alpha .

$$\beta =1-\frac{2}{3}\times \frac{9}{10}$$

Calculate \beta from \beta =1-\frac{2}{3}\times \frac{9}{10}.

$$\beta =\frac{2}{5}$$

Substitute \frac{2}{5} for \beta and \frac{9}{10} for \alpha in the equation t=\frac{2}{3}\alpha +\frac{1}{4}\beta .

$$t=\frac{2}{3}\times \frac{9}{10}+\frac{1}{4}\times \frac{2}{5}$$

Calculate t from t=\frac{2}{3}\times \frac{9}{10}+\frac{1}{4}\times \frac{2}{5}.

$$t=\frac{7}{10}$$

The system is now solved.

$$\alpha =\frac{9}{10} \beta =\frac{2}{5} t=\frac{7}{10}$$