Solve {l}{4/x+5/4=-2}{3/x-5/y=21}

$$\left\{ \begin{array} { l } { \frac { 4 } { x } + \frac { 5 } { 4 } = - 2 } \\ { \frac { 3 } { x } - \frac { 5 } { y } = 21 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{16}{13} = -1\frac{3}{13} \approx -1.230769231$$

$$y=-\frac{16}{75}\approx -0.213333333$$

Solution Steps

Consider the first equation. 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.

$$4\times 4+4x\times \frac{5}{4}=-8x$$

Multiply 4 and 4 to get 16.

$$16+4x\times \frac{5}{4}=-8x$$

Multiply 4 and \frac{5}{4} to get 5.

$$16+5x=-8x$$

Add 8x to both sides.

$$16+5x+8x=0$$

Combine 5x and 8x to get 13x.

$$16+13x=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$13x=-16$$

Divide both sides by 13.

$$x=-\frac{16}{13}$$

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

$$\frac{3}{-\frac{16}{13}}-\frac{5}{y}=21$$

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

$$y\times \frac{3}{-\frac{16}{13}}-5=21y$$

Divide 3 by -\frac{16}{13} by multiplying 3 by the reciprocal of -\frac{16}{13}.

$$y\times 3\left(-\frac{13}{16}\right)-5=21y$$

Multiply 3 and -\frac{13}{16} to get -\frac{39}{16}.

$$y\left(-\frac{39}{16}\right)-5=21y$$

Subtract 21y from both sides.

$$y\left(-\frac{39}{16}\right)-5-21y=0$$

Combine y\left(-\frac{39}{16}\right) and -21y to get -\frac{375}{16}y.

$$-\frac{375}{16}y-5=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$-\frac{375}{16}y=5$$

Multiply both sides by -\frac{16}{375}, the reciprocal of -\frac{375}{16}.

$$y=5\left(-\frac{16}{375}\right)$$

Multiply 5 and -\frac{16}{375} to get -\frac{16}{75}.

$$y=-\frac{16}{75}$$

The system is now solved.

$$x=-\frac{16}{13} y=-\frac{16}{75}$$