Solve {l}{3x=30}{x+2y=20}{y+2z=9}{k=y+1/2z+(x-1/2z)}

$$\left\{ \begin{array} { l } { 3 x = 30 } \\ { x + 2 y = 20 } \\ { y + 2 z = 9 } \\ { k = y + \frac { 1 } { 2 } z + ( x - \frac { 1 } { 2 } z ) } \end{array} \right.$$

Solve for x, y, z, k

$$x=10$$

$$y=5$$

$$z=2$$

$$k=15$$

Solution Steps

Consider the first equation. Divide both sides by 3.

$$x=\frac{30}{3}$$

Divide 30 by 3 to get 10.

$$x=10$$

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

$$10+2y=20$$

Subtract 10 from both sides.

$$2y=20-10$$

Subtract 10 from 20 to get 10.

$$2y=10$$

Divide both sides by 2.

$$y=\frac{10}{2}$$

Divide 10 by 2 to get 5.

$$y=5$$

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

$$5+2z=9$$

Subtract 5 from both sides.

$$2z=9-5$$

Subtract 5 from 9 to get 4.

$$2z=4$$

Divide both sides by 2.

$$z=\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$z=2$$

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

$$k=5+\frac{1}{2}\times 2+10-\frac{1}{2}\times 2$$

Multiply \frac{1}{2} and 2 to get 1.

$$k=5+1+10-\frac{1}{2}\times 2$$

Add 5 and 1 to get 6.

$$k=6+10-\frac{1}{2}\times 2$$

Add 6 and 10 to get 16.

$$k=16-\frac{1}{2}\times 2$$

Multiply -\frac{1}{2} and 2 to get -1.

$$k=16-1$$

Subtract 1 from 16 to get 15.

$$k=15$$

The system is now solved.

$$x=10 y=5 z=2 k=15$$