Solve {l}{1/x+4/y=1/15}{1/x+frac{2}{5}=frac{1}{24}}

$$\left. \begin{array} { l } { \frac { 1 } { x } + \frac { 4 } { y } = \frac { 1 } { 15 } } \\ { \frac { 1 } { x } + \frac { 2 } { 5 } = \frac { 1 } { 24 } } \end{array} \right.$$

Solve for x, y

$$x = -\frac{120}{43} = -2\frac{34}{43} \approx -2.790697674$$

$$y = \frac{160}{17} = 9\frac{7}{17} \approx 9.411764706$$

Solution Steps

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 120x, the least common multiple of x,5,24.

$$120+120x\times \frac{2}{5}=5x$$

Multiply 120 and \frac{2}{5} to get 48.

$$120+48x=5x$$

Subtract 5x from both sides.

$$120+48x-5x=0$$

Combine 48x and -5x to get 43x.

$$120+43x=0$$

Subtract 120 from both sides. Anything subtracted from zero gives its negation.

$$43x=-120$$

Divide both sides by 43.

$$x=-\frac{120}{43}$$

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

$$\frac{1}{-\frac{120}{43}}+\frac{4}{y}=\frac{1}{15}$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 15y, the least common multiple of y,15.

$$15y\times \frac{1}{-\frac{120}{43}}+15\times 4=y$$

Divide 1 by -\frac{120}{43} by multiplying 1 by the reciprocal of -\frac{120}{43}.

$$15y\times 1\left(-\frac{43}{120}\right)+15\times 4=y$$

Multiply 1 and -\frac{43}{120} to get -\frac{43}{120}.

$$15y\left(-\frac{43}{120}\right)+15\times 4=y$$

Multiply 15 and -\frac{43}{120} to get -\frac{43}{8}.

$$-\frac{43}{8}y+15\times 4=y$$

Multiply 15 and 4 to get 60.

$$-\frac{43}{8}y+60=y$$

Subtract y from both sides.

$$-\frac{43}{8}y+60-y=0$$

Combine -\frac{43}{8}y and -y to get -\frac{51}{8}y.

$$-\frac{51}{8}y+60=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{51}{8}y=-60$$

Multiply both sides by -\frac{8}{51}, the reciprocal of -\frac{51}{8}.

$$y=-60\left(-\frac{8}{51}\right)$$

Multiply -60 and -\frac{8}{51} to get \frac{160}{17}.

$$y=\frac{160}{17}$$

The system is now solved.

$$x=-\frac{120}{43} y=\frac{160}{17}$$