Solve {l}{x+2y=6+x}{2x-y=-2k}{3x+y=5}

$$\left. \begin{array} { l } { x + 2 y = 6 + x } \\ { 2 x - y = - 2 k } \\ { 3 x + y = 5 } \end{array} \right.$$

Solve for x, y, k

$$x=\frac{2}{3}\approx 0.666666667$$

$$y=3$$

$$k=\frac{5}{6}\approx 0.833333333$$

Solution Steps

Consider the first equation. Subtract x from both sides.

$$x+2y-x=6$$

Combine x and -x to get 0.

$$2y=6$$

Divide both sides by 2.

$$y=\frac{6}{2}$$

Divide 6 by 2 to get 3.

$$y=3$$

Consider the second equation. Insert the known values of variables into the equation.

$$2x-3=-2k$$

Add 2k to both sides.

$$2x-3+2k=0$$

Add 3 to both sides. Anything plus zero gives itself.

$$2x+2k=3$$

Consider the third equation. Insert the known values of variables into the equation.

$$3x+3=5$$

Subtract 3 from both sides.

$$3x=5-3$$

Subtract 3 from 5 to get 2.

$$3x=2$$

Divide both sides by 3.

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

Consider the second equation. Insert the known values of variables into the equation.

$$2\times \frac{2}{3}+2k=3$$

Multiply 2 and \frac{2}{3} to get \frac{4}{3}.

$$\frac{4}{3}+2k=3$$

Subtract \frac{4}{3} from both sides.

$$2k=3-\frac{4}{3}$$

Subtract \frac{4}{3} from 3 to get \frac{5}{3}.

$$2k=\frac{5}{3}$$

Divide both sides by 2.

$$k=\frac{\frac{5}{3}}{2}$$

Express \frac{\frac{5}{3}}{2} as a single fraction.

$$k=\frac{5}{3\times 2}$$

Multiply 3 and 2 to get 6.

$$k=\frac{5}{6}$$

The system is now solved.

$$x=\frac{2}{3} y=3 k=\frac{5}{6}$$