Solve {l}{2/x+3/y+10/2=4}{4/2-frac{6}{y}+frac{5}{2}=1}{frac{6}{x}+frac{9}{y}-frac{20}{z}=2}

$$\left. \begin{array} { l } { \frac { 2 } { x } + \frac { 3 } { y } + \frac { 10 } { 2 } = 4 } \\ { \frac { 4 } { 2 } - \frac { 6 } { y } + \frac { 5 } { 2 } = 1 } \\ { \frac { 6 } { x } + \frac { 9 } { y } - \frac { 20 } { z } = 2 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{8}{11}\approx -0.727272727$$

$$y = \frac{12}{7} = 1\frac{5}{7} \approx 1.714285714$$

$$z=\frac{80}{24t_{1744830465}+13}$$

$$t_{1744830465}\neq -\frac{13}{24}$$

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 2,y.

$$y\times 4-2\times 6+2y\times \frac{5}{2}=2y$$

Multiply -2 and 6 to get -12.

$$y\times 4-12+2y\times \frac{5}{2}=2y$$

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

$$y\times 4-12+5y=2y$$

Combine y\times 4 and 5y to get 9y.

$$9y-12=2y$$

Subtract 2y from both sides.

$$9y-12-2y=0$$

Combine 9y and -2y to get 7y.

$$7y-12=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$7y=12$$

Divide both sides by 7.

$$y=\frac{12}{7}$$

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

$$\frac{2}{x}+\frac{3}{\frac{12}{7}}+\frac{10}{2}=4$$

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.

$$2\times 2+2x\times \frac{3}{\frac{12}{7}}+x\times 10=8x$$

Multiply 2 and 2 to get 4.

$$4+2x\times \frac{3}{\frac{12}{7}}+x\times 10=8x$$

Divide 3 by \frac{12}{7} by multiplying 3 by the reciprocal of \frac{12}{7}.

$$4+2x\times 3\times \frac{7}{12}+x\times 10=8x$$

Multiply 3 and \frac{7}{12} to get \frac{7}{4}.

$$4+2x\times \frac{7}{4}+x\times 10=8x$$

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

$$4+\frac{7}{2}x+x\times 10=8x$$

Combine \frac{7}{2}x and x\times 10 to get \frac{27}{2}x.

$$4+\frac{27}{2}x=8x$$

Subtract 8x from both sides.

$$4+\frac{27}{2}x-8x=0$$

Combine \frac{27}{2}x and -8x to get \frac{11}{2}x.

$$4+\frac{11}{2}x=0$$

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

$$\frac{11}{2}x=-4$$

Multiply both sides by \frac{2}{11}, the reciprocal of \frac{11}{2}.

$$x=-4\times \frac{2}{11}$$

Multiply -4 and \frac{2}{11} to get -\frac{8}{11}.

$$x=-\frac{8}{11}$$

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

$$\frac{6}{-\frac{8}{11}}+\frac{9}{\frac{12}{7}}-\frac{20}{z}=2$$

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

$$z\times \frac{6}{-\frac{8}{11}}+z\times \frac{9}{\frac{12}{7}}-20=2z$$

Divide 6 by -\frac{8}{11} by multiplying 6 by the reciprocal of -\frac{8}{11}.

$$z\times 6\left(-\frac{11}{8}\right)+z\times \frac{9}{\frac{12}{7}}-20=2z$$

Multiply 6 and -\frac{11}{8} to get -\frac{33}{4}.

$$z\left(-\frac{33}{4}\right)+z\times \frac{9}{\frac{12}{7}}-20=2z$$

Divide 9 by \frac{12}{7} by multiplying 9 by the reciprocal of \frac{12}{7}.

$$z\left(-\frac{33}{4}\right)+z\times 9\times \frac{7}{12}-20=2z$$

Multiply 9 and \frac{7}{12} to get \frac{21}{4}.

$$z\left(-\frac{33}{4}\right)+z\times \frac{21}{4}-20=2z$$

Combine z\left(-\frac{33}{4}\right) and z\times \frac{21}{4} to get -3z.

$$-3z-20=2z$$

Subtract 2z from both sides.

$$-3z-20-2z=0$$

Combine -3z and -2z to get -5z.

$$-5z-20=0$$

Add 20 to both sides. Anything plus zero gives itself.

$$-5z=20$$

Divide both sides by -5.

$$z=\frac{20}{-5}$$

Divide 20 by -5 to get -4.

$$z=-4$$

The system is now solved.

$$x=-\frac{8}{11} y=\frac{12}{7} z=-4$$