Solve {c}{2x+2mu=1}{2lambda-4mu=6}{lambda+mu=0}

$$\left. \begin{array} { c } { 2 x + 2 \mu = 1 } \\ { 2 \lambda - 4 \mu = 6 } \\ { \lambda + \mu = 0 } \end{array} \right.$$

Solve for x, μ, λ

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$\mu =-1$$

$$\lambda =1$$

Short Steps Using Substitution

Reorder the equations.

$$\lambda +\mu =0 2\lambda -4\mu =6 2x+2\mu =1$$

Solve \lambda +\mu =0 for \mu .

$$\mu =-\lambda$$

Substitute -\lambda for \mu in the second and third equation.

$$2\lambda -4\left(-1\right)\lambda =6 2x+2\left(-1\right)\lambda =1$$

Solve these equations for \lambda and x respectively.

$$\lambda =1 x=\lambda +\frac{1}{2}$$

Substitute 1 for \lambda in the equation x=\lambda +\frac{1}{2}.

$$x=1+\frac{1}{2}$$

Calculate x from x=1+\frac{1}{2}.

$$x=\frac{3}{2}$$

Substitute 1 for \lambda in the equation \mu =-\lambda .

$$\mu =-1$$

The system is now solved.

$$x=\frac{3}{2} \mu =-1 \lambda =1$$