Solve {l}{{({(frac{3*{(3)}+1}{3})}*9/10+x/5)}*{(frac{3*{(7)}+1}{7})}-{(frac{15*{(10)}+3}{10})}={(frac{6*{(10)}+7}{10})}*1{(frac{3*{(3)}+1}{3})}-{(frac{6*{(10)}+7}{10})}*{(frac{12*{(3)}+1}{3})}}{y=x}{text{Solvefor}ztext{where}}{z=y}

$$\left. \begin{array} { l } { {({(\frac{3 \cdot {(3)} + 1}{3})} \cdot \frac{9}{10} + x / 5)} \cdot {(\frac{3 \cdot {(7)} + 1}{7})} - {(\frac{15 \cdot {(10)} + 3}{10})} = {(\frac{6 \cdot {(10)} + 7}{10})} \cdot 1 {(\frac{3 \cdot {(3)} + 1}{3})} - {(\frac{6 \cdot {(10)} + 7}{10})} \cdot {(\frac{12 \cdot {(3)} + 1}{3})} }\\ { y = x }\\ { \text{Solve for } z \text{ where} } \\ { z = y } \end{array} \right.$$

Solve for x, y, z

$$z = -\frac{1905}{22} = -86\frac{13}{22} \approx -86.590909091$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 210, the least common multiple of 3,10,5,7.

$$30\left(\frac{3\times 3+1}{3}\times \frac{9}{10}+\frac{x}{5}\right)\left(3\times 7+1\right)-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 3 and 3 to get 9.

$$30\left(\frac{9+1}{3}\times \frac{9}{10}+\frac{x}{5}\right)\left(3\times 7+1\right)-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Add 9 and 1 to get 10.

$$30\left(\frac{10}{3}\times \frac{9}{10}+\frac{x}{5}\right)\left(3\times 7+1\right)-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply \frac{10}{3} and \frac{9}{10} to get 3.

$$30\left(3+\frac{x}{5}\right)\left(3\times 7+1\right)-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 3 and 7 to get 21.

$$30\left(3+\frac{x}{5}\right)\left(21+1\right)-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Add 21 and 1 to get 22.

$$30\left(3+\frac{x}{5}\right)\times 22-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 30 and 22 to get 660.

$$660\left(3+\frac{x}{5}\right)-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Use the distributive property to multiply 660 by 3+\frac{x}{5}.

$$1980+660\times \frac{x}{5}-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Cancel out 5, the greatest common factor in 660 and 5.

$$1980+132x-21\left(15\times 10+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 15 and 10 to get 150.

$$1980+132x-21\left(150+3\right)=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Add 150 and 3 to get 153.

$$1980+132x-21\times 153=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply -21 and 153 to get -3213.

$$1980+132x-3213=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Subtract 3213 from 1980 to get -1233.

$$-1233+132x=7\left(6\times 10+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 6 and 10 to get 60.

$$-1233+132x=7\left(60+7\right)\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Add 60 and 7 to get 67.

$$-1233+132x=7\times 67\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 7 and 67 to get 469.

$$-1233+132x=469\times 1\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 469 and 1 to get 469.

$$-1233+132x=469\left(3\times 3+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 3 and 3 to get 9.

$$-1233+132x=469\left(9+1\right)-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Add 9 and 1 to get 10.

$$-1233+132x=469\times 10-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 469 and 10 to get 4690.

$$-1233+132x=4690-7\left(6\times 10+7\right)\left(12\times 3+1\right)$$

Multiply 6 and 10 to get 60.

$$-1233+132x=4690-7\left(60+7\right)\left(12\times 3+1\right)$$

Add 60 and 7 to get 67.

$$-1233+132x=4690-7\times 67\left(12\times 3+1\right)$$

Multiply 7 and 67 to get 469.

$$-1233+132x=4690-469\left(12\times 3+1\right)$$

Multiply 12 and 3 to get 36.

$$-1233+132x=4690-469\left(36+1\right)$$

Add 36 and 1 to get 37.

$$-1233+132x=4690-469\times 37$$

Multiply 469 and 37 to get 17353.

$$-1233+132x=4690-17353$$

Subtract 17353 from 4690 to get -12663.

$$-1233+132x=-12663$$

Add 1233 to both sides.

$$132x=-12663+1233$$

Add -12663 and 1233 to get -11430.

$$132x=-11430$$

Divide both sides by 132.

$$x=\frac{-11430}{132}$$

Reduce the fraction \frac{-11430}{132} to lowest terms by extracting and canceling out 6.

$$x=-\frac{1905}{22}$$

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

$$y=-\frac{1905}{22}$$

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

$$z=-\frac{1905}{22}$$

The system is now solved.

$$x=-\frac{1905}{22} y=-\frac{1905}{22} z=-\frac{1905}{22}$$