Solve {l}{125Q=250-25Q}{P=250-12*5Q}{M=1.25Q}{A=500/Q+125Q}{T=500+.625Q^2}

$$\left. \begin{array} { l } { 125 Q = 250 - 25 Q } \\ { P = 250 - 12 \cdot 5 Q } \\ { M = 1.25 Q } \\ { A = \frac { 500 } { Q } + 125 Q } \\ { T = 500 + .625 Q ^ { 2 } } \end{array} \right.$$

Solve for Q, P, M, A, T

$$Q = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$P=150$$

$$M = \frac{25}{12} = 2\frac{1}{12} \approx 2.083333333$$

$$A = \frac{1525}{3} = 508\frac{1}{3} \approx 508.333333333$$

$$T = \frac{36125}{72} = 501\frac{53}{72} \approx 501.736111111$$

Solution Steps

Consider the first equation. Add 25Q to both sides.

$$125Q+25Q=250$$

Combine 125Q and 25Q to get 150Q.

$$150Q=250$$

Divide both sides by 150.

$$Q=\frac{250}{150}$$

Reduce the fraction \frac{250}{150} to lowest terms by extracting and canceling out 50.

$$Q=\frac{5}{3}$$

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

$$P=250-12\times 5\times \frac{5}{3}$$

Multiply 12 and 5 to get 60.

$$P=250-60\times \frac{5}{3}$$

Multiply 60 and \frac{5}{3} to get 100.

$$P=250-100$$

Subtract 100 from 250 to get 150.

$$P=150$$

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

$$M=1.25\times \frac{5}{3}$$

Multiply 1.25 and \frac{5}{3} to get \frac{25}{12}.

$$M=\frac{25}{12}$$

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

$$A=\frac{500}{\frac{5}{3}}+125\times \frac{5}{3}$$

Divide 500 by \frac{5}{3} by multiplying 500 by the reciprocal of \frac{5}{3}.

$$A=500\times \frac{3}{5}+125\times \frac{5}{3}$$

Multiply 500 and \frac{3}{5} to get 300.

$$A=300+125\times \frac{5}{3}$$

Multiply 125 and \frac{5}{3} to get \frac{625}{3}.

$$A=300+\frac{625}{3}$$

Add 300 and \frac{625}{3} to get \frac{1525}{3}.

$$A=\frac{1525}{3}$$

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

$$T=500+0.625\times \left(\frac{5}{3}\right)^{2}$$

Calculate \frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$T=500+0.625\times \frac{25}{9}$$

Multiply 0.625 and \frac{25}{9} to get \frac{125}{72}.

$$T=500+\frac{125}{72}$$

Add 500 and \frac{125}{72} to get \frac{36125}{72}.

$$T=\frac{36125}{72}$$

The system is now solved.

$$Q=\frac{5}{3} P=150 M=\frac{25}{12} A=\frac{1525}{3} T=\frac{36125}{72}$$