Solve {l}{x/3+4/2-z=7}{x/4-3y/2+frac{z}{2}=-6}{frac{x}{6}-frac{4}{4}-frac{2}{3}=1}

$$\left. \begin{array} { l } { \frac { x } { 3 } + \frac { 4 } { 2 } - z = 7 } \\ { \frac { x } { 4 } - \frac { 3 y } { 2 } + \frac { z } { 2 } = - 6 } \\ { \frac { x } { 6 } - \frac { 4 } { 4 } - \frac { 2 } { 3 } = 1 } \end{array} \right.$$

Solve for x, z, y

$$x=16$$

$$y = \frac{61}{9} = 6\frac{7}{9} \approx 6.777777778$$

$$z=\frac{1}{3}\approx 0.333333333$$

Solution Steps

Consider the third equation. Multiply both sides of the equation by 12, the least common multiple of 6,4,3.

$$2x-3\times 4-8=12$$

Multiply -3 and 4 to get -12.

$$2x-12-8=12$$

Subtract 8 from -12 to get -20.

$$2x-20=12$$

Add 20 to both sides.

$$2x=12+20$$

Add 12 and 20 to get 32.

$$2x=32$$

Divide both sides by 2.

$$x=\frac{32}{2}$$

Divide 32 by 2 to get 16.

$$x=16$$

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

$$\frac{16}{3}+\frac{4}{2}-z=7$$

Divide 4 by 2 to get 2.

$$\frac{16}{3}+2-z=7$$

Add \frac{16}{3} and 2 to get \frac{22}{3}.

$$\frac{22}{3}-z=7$$

Subtract \frac{22}{3} from both sides.

$$-z=7-\frac{22}{3}$$

Subtract \frac{22}{3} from 7 to get -\frac{1}{3}.

$$-z=-\frac{1}{3}$$

Divide both sides by -1.

$$z=\frac{-\frac{1}{3}}{-1}$$

Express \frac{-\frac{1}{3}}{-1} as a single fraction.

$$z=\frac{-1}{3\left(-1\right)}$$

Cancel out -1 in both numerator and denominator.

$$z=\frac{1}{3}$$

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

$$\frac{16}{4}-\frac{3y}{2}+\frac{\frac{1}{3}}{2}=-6$$

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$16-2\times 3y+2\times \frac{1}{3}=-24$$

Multiply -2 and 3 to get -6.

$$16-6y+2\times \frac{1}{3}=-24$$

Multiply 2 and \frac{1}{3} to get \frac{2}{3}.

$$16-6y+\frac{2}{3}=-24$$

Add 16 and \frac{2}{3} to get \frac{50}{3}.

$$\frac{50}{3}-6y=-24$$

Subtract \frac{50}{3} from both sides.

$$-6y=-24-\frac{50}{3}$$

Subtract \frac{50}{3} from -24 to get -\frac{122}{3}.

$$-6y=-\frac{122}{3}$$

Divide both sides by -6.

$$y=\frac{-\frac{122}{3}}{-6}$$

Express \frac{-\frac{122}{3}}{-6} as a single fraction.

$$y=\frac{-122}{3\left(-6\right)}$$

Multiply 3 and -6 to get -18.

$$y=\frac{-122}{-18}$$

Reduce the fraction \frac{-122}{-18} to lowest terms by extracting and canceling out -2.

$$y=\frac{61}{9}$$

The system is now solved.

$$x=16 z=\frac{1}{3} y=\frac{61}{9}$$