Solve (1)/(x)+(1)/(y)-(1)/(z)=5 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Solve for x

$x=-\frac{yz}{z-y-5yz}$
$y\neq 0\text{ and }z\neq 0\text{ and }\left(z=-\frac{1}{5}\text{ or }y\neq \frac{z}{5z+1}\right)$

Steps for Solving Linear Equation

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

$$yz+xz-xy=5xyz$$

Subtract $5xyz$ from both sides.

$$yz+xz-xy-5xyz=0$$

Subtract $yz$ from both sides. Anything subtracted from zero gives its negation.

$$xz-xy-5xyz=-yz$$

Reorder the terms.

$$-5xyz-xy+xz=-yz$$

Combine all terms containing $x$.

$$\left(-5yz-y+z\right)x=-yz$$

The equation is in standard form.

$$\left(z-y-5yz\right)x=-yz$$

Divide both sides by $z-y-5zy$.

$$\frac{\left(z-y-5yz\right)x}{z-y-5yz}=-\frac{yz}{z-y-5yz}$$

Dividing by $z-y-5zy$ undoes the multiplication by $z-y-5zy$.

$$x=-\frac{yz}{z-y-5yz}$$

Variable $x$ cannot be equal to $0$.

$$x=-\frac{yz}{z-y-5yz}\text{, }x\neq 0$$

Solve for y

$y=-\frac{xz}{z-x-5xz}$
$z\neq 0\text{ and }x\neq 0\text{ and }\left(z=-\frac{1}{5}\text{ or }x\neq \frac{z}{5z+1}\right)$

Steps for Solving Linear Equation

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

$$yz+xz-xy=5xyz$$

Subtract $5xyz$ from both sides.

$$yz+xz-xy-5xyz=0$$

Subtract $xz$ from both sides. Anything subtracted from zero gives its negation.

$$yz-xy-5xyz=-xz$$

Reorder the terms.

$$-5xyz-xy+yz=-xz$$

Combine all terms containing $y$.

$$\left(-5xz-x+z\right)y=-xz$$

The equation is in standard form.

$$\left(z-x-5xz\right)y=-xz$$

Divide both sides by $z-x-5zx$.

$$\frac{\left(z-x-5xz\right)y}{z-x-5xz}=-\frac{xz}{z-x-5xz}$$

Dividing by $z-x-5zx$ undoes the multiplication by $z-x-5zx$.

$$y=-\frac{xz}{z-x-5xz}$$

Variable $y$ cannot be equal to $0$.

$$y=-\frac{xz}{z-x-5xz}\text{, }y\neq 0$$