Solve {l}{x+3y=5k+1}{2x-5y=13-k}{3x-2y=1/2k+1}

$$\left\{ \begin{array} { l } { x + 3 y = 5 k + 1 } \\ { 2 x - 5 y = 13 - k } \\ { 3 x - 2 y = \frac { 1 } { 2 } k + 1 } \end{array} \right.$$

Solve for x, y, k

$$x = -\frac{24}{7} = -3\frac{3}{7} \approx -3.428571429$$

$$y = -\frac{33}{7} = -4\frac{5}{7} \approx -4.714285714$$

$$k = -\frac{26}{7} = -3\frac{5}{7} \approx -3.714285714$$

Short Steps Using Substitution

Solve x+3y=5k+1 for x.

$$x=-3y+5k+1$$

Substitute -3y+5k+1 for x in the second and third equation.

$$2\left(-3y+5k+1\right)-5y=13-k 3\left(-3y+5k+1\right)-2y=\frac{1}{2}k+1$$

Solve these equations for y and k respectively.

$$y=-1+k k=-\frac{4}{29}+\frac{22}{29}y$$

Substitute -1+k for y in the equation k=-\frac{4}{29}+\frac{22}{29}y.

$$k=-\frac{4}{29}+\frac{22}{29}\left(-1+k\right)$$

Solve k=-\frac{4}{29}+\frac{22}{29}\left(-1+k\right) for k.

$$k=-\frac{26}{7}$$

Substitute -\frac{26}{7} for k in the equation y=-1+k.

$$y=-1-\frac{26}{7}$$

Calculate y from y=-1-\frac{26}{7}.

$$y=-\frac{33}{7}$$

Substitute -\frac{33}{7} for y and -\frac{26}{7} for k in the equation x=-3y+5k+1.

$$x=-3\left(-\frac{33}{7}\right)+5\left(-\frac{26}{7}\right)+1$$

Calculate x from x=-3\left(-\frac{33}{7}\right)+5\left(-\frac{26}{7}\right)+1.

$$x=-\frac{24}{7}$$

The system is now solved.

$$x=-\frac{24}{7} y=-\frac{33}{7} k=-\frac{26}{7}$$