Solve {l}{1+1/y=1/8}{1/x+4/y=frac{1}{4}}

$$\left\{ \begin{array} { l } { 1 + \frac { 1 } { y } = \frac { 1 } { 8 } } \\ { \frac { 1 } { x } + \frac { 4 } { y } = \frac { 1 } { 4 } } \end{array} \right.$$

Solve for y, x

$$x=\frac{4}{15}\approx 0.266666667$$

$$y = -\frac{8}{7} = -1\frac{1}{7} \approx -1.142857143$$

Solution Steps

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8y, the least common multiple of y,8.

$$8y+8=y$$

Subtract y from both sides.

$$8y+8-y=0$$

Combine 8y and -y to get 7y.

$$7y+8=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$7y=-8$$

Divide both sides by 7.

$$y=-\frac{8}{7}$$

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

$$\frac{1}{x}+\frac{4}{-\frac{8}{7}}=\frac{1}{4}$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x, the least common multiple of x,4.

$$4+4x\times \frac{4}{-\frac{8}{7}}=x$$

Divide 4 by -\frac{8}{7} by multiplying 4 by the reciprocal of -\frac{8}{7}.

$$4+4x\times 4\left(-\frac{7}{8}\right)=x$$

Multiply 4 and -\frac{7}{8} to get -\frac{7}{2}.

$$4+4x\left(-\frac{7}{2}\right)=x$$

Multiply 4 and -\frac{7}{2} to get -14.

$$4-14x=x$$

Subtract x from both sides.

$$4-14x-x=0$$

Combine -14x and -x to get -15x.

$$4-15x=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-15x=-4$$

Divide both sides by -15.

$$x=\frac{-4}{-15}$$

Fraction \frac{-4}{-15} can be simplified to \frac{4}{15} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{4}{15}$$

The system is now solved.

$$y=-\frac{8}{7} x=\frac{4}{15}$$