Solve 1/x-2/y+4=0;1/y-1/2+1=0;frac{2}{z}+frac{3}{x}=14

$$\frac { 1 } { x } - \frac { 2 } { y } + 4 = 0 ; \frac { 1 } { y } - \frac { 1 } { 2 } + 1 = 0 ; \frac { 2 } { z } + \frac { 3 } { x } = 14$$

Solve for x, y, z

$$x=-\frac{1}{5}=-0.2$$

$$y=-2$$

$$z=\frac{2}{29}\approx 0.068965517$$

Solution Steps

Consider the second equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2y, the least common multiple of y,2.

$$2+2y\left(-\frac{1}{2}\right)+2y=0$$

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

$$2-y+2y=0$$

Combine -y and 2y to get y.

$$2+y=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$y=-2$$

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

$$\frac{1}{x}-\frac{2}{-2}+4=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of x,-2.

$$2-\left(-x\times 2\right)+2x\times 4=0$$

Multiply -1 and 2 to get -2.

$$2-\left(-2x\right)+2x\times 4=0$$

The opposite of -2x is 2x.

$$2+2x+2x\times 4=0$$

Multiply 2 and 4 to get 8.

$$2+2x+8x=0$$

Combine 2x and 8x to get 10x.

$$2+10x=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$10x=-2$$

Divide both sides by 10.

$$x=\frac{-2}{10}$$

Reduce the fraction \frac{-2}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{5}$$

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

$$\frac{2}{z}+\frac{3}{-\frac{1}{5}}=14$$

Variable z cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by z.

$$2+z\times \frac{3}{-\frac{1}{5}}=14z$$

Divide 3 by -\frac{1}{5} by multiplying 3 by the reciprocal of -\frac{1}{5}.

$$2+z\times 3\left(-5\right)=14z$$

Multiply 3 and -5 to get -15.

$$2+z\left(-15\right)=14z$$

Subtract 14z from both sides.

$$2+z\left(-15\right)-14z=0$$

Combine z\left(-15\right) and -14z to get -29z.

$$2-29z=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-29z=-2$$

Divide both sides by -29.

$$z=\frac{-2}{-29}$$

Fraction \frac{-2}{-29} can be simplified to \frac{2}{29} by removing the negative sign from both the numerator and the denominator.

$$z=\frac{2}{29}$$

The system is now solved.

$$x=-\frac{1}{5} y=-2 z=\frac{2}{29}$$