Solve {l}{0=4-4t+3s}{0=3t+5}{z=-2-t-3s}

$$\left\{ \begin{array} { l } { 0 = 4 - 4 t + 3 s } \\ { 0 = 3 t + 5 } \\ { z = - 2 - t - 3 s } \end{array} \right.$$

Solve for t, s, z

$$z = \frac{31}{3} = 10\frac{1}{3} \approx 10.333333333$$

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

$$s = -\frac{32}{9} = -3\frac{5}{9} \approx -3.555555556$$

Solution Steps

Consider the second equation. Swap sides so that all variable terms are on the left hand side.

$$3t+5=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$3t=-5$$

Divide both sides by 3.

$$t=-\frac{5}{3}$$

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

$$0=4-4\left(-\frac{5}{3}\right)+3s$$

Multiply -4 and -\frac{5}{3} to get \frac{20}{3}.

$$0=4+\frac{20}{3}+3s$$

Add 4 and \frac{20}{3} to get \frac{32}{3}.

$$0=\frac{32}{3}+3s$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{32}{3}+3s=0$$

Subtract \frac{32}{3} from both sides. Anything subtracted from zero gives its negation.

$$3s=-\frac{32}{3}$$

Divide both sides by 3.

$$s=\frac{-\frac{32}{3}}{3}$$

Express \frac{-\frac{32}{3}}{3} as a single fraction.

$$s=\frac{-32}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$s=\frac{-32}{9}$$

Fraction \frac{-32}{9} can be rewritten as -\frac{32}{9} by extracting the negative sign.

$$s=-\frac{32}{9}$$

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

$$z=-2-\left(-\frac{5}{3}\right)-3\left(-\frac{32}{9}\right)$$

Multiply -1 and -\frac{5}{3} to get \frac{5}{3}.

$$z=-2+\frac{5}{3}-3\left(-\frac{32}{9}\right)$$

Add -2 and \frac{5}{3} to get -\frac{1}{3}.

$$z=-\frac{1}{3}-3\left(-\frac{32}{9}\right)$$

Multiply -3 and -\frac{32}{9} to get \frac{32}{3}.

$$z=-\frac{1}{3}+\frac{32}{3}$$

Add -\frac{1}{3} and \frac{32}{3} to get \frac{31}{3}.

$$z=\frac{31}{3}$$

The system is now solved.

$$t=-\frac{5}{3} s=-\frac{32}{9} z=\frac{31}{3}$$