Solve {l}{8x-2y+lambda=0}{-2x+12y+lambda=0}{x+y-72=0}

$$\left. \begin{array} { l } { 8 x - 2 y + \lambda = 0 } \\ { - 2 x + 12 y + \lambda = 0 } \\ { x + y - 72 = 0 } \end{array} \right.$$

Solve for x, y, λ

$$x=42$$

$$y=30$$

$$\lambda =-276$$

Short Steps Using Substitution

Solve 8x-2y+\lambda =0 for \lambda .

$$\lambda =-8x+2y$$

Substitute -8x+2y for \lambda in the equation -2x+12y+\lambda =0.

$$-2x+12y-8x+2y=0$$

Solve the second equation for y and the third equation for x.

$$y=\frac{5}{7}x x=-y+72$$

Substitute \frac{5}{7}x for y in the equation x=-y+72.

$$x=-\frac{5}{7}x+72$$

Solve x=-\frac{5}{7}x+72 for x.

$$x=42$$

Substitute 42 for x in the equation y=\frac{5}{7}x.

$$y=\frac{5}{7}\times 42$$

Calculate y from y=\frac{5}{7}\times 42.

$$y=30$$

Substitute 30 for y and 42 for x in the equation \lambda =-8x+2y.

$$\lambda =-8\times 42+2\times 30$$

Calculate \lambda from \lambda =-8\times 42+2\times 30.

$$\lambda =-276$$

The system is now solved.

$$x=42 y=30 \lambda =-276$$