Solve {l}{g{(n)}=3n-4}{h=g{(-10)}}{text{Solvefor}itext{where}}{i=h}

$$\left. \begin{array} { l } { g {(n)} = 3 n - 4 }\\ { h = g {(-10)} }\\ { \text{Solve for } i \text{ where} } \\ { i = h } \end{array} \right.$$

Solve for g, n, h

$$g=-\frac{1}{10}i=-0.1i$$

$$n=\frac{1200}{901}-\frac{40}{901}i\approx 1.331853496-0.044395117i$$

$$h=i$$

Solution Steps

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

$$h=i$$

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

$$i=g\left(-10\right)$$

Divide both sides by -10.

$$\frac{i}{-10}=g$$

Divide i by -10 to get -\frac{1}{10}i.

$$-\frac{1}{10}i=g$$

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

$$g=-\frac{1}{10}i$$

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

$$-\frac{1}{10}in=3n-4$$

Subtract 3n from both sides.

$$-\frac{1}{10}in-3n=-4$$

Combine -\frac{1}{10}in and -3n to get \left(-3-\frac{1}{10}i\right)n.

$$\left(-3-\frac{1}{10}i\right)n=-4$$

Divide both sides by -3-\frac{1}{10}i.

$$n=\frac{-4}{-3-\frac{1}{10}i}$$

Multiply both numerator and denominator of \frac{-4}{-3-\frac{1}{10}i} by the complex conjugate of the denominator, -3+\frac{1}{10}i.

$$n=\frac{-4\left(-3+\frac{1}{10}i\right)}{\left(-3-\frac{1}{10}i\right)\left(-3+\frac{1}{10}i\right)}$$

Do the multiplications in \frac{-4\left(-3+\frac{1}{10}i\right)}{\left(-3-\frac{1}{10}i\right)\left(-3+\frac{1}{10}i\right)}.

$$n=\frac{12-\frac{2}{5}i}{\frac{901}{100}}$$

Divide 12-\frac{2}{5}i by \frac{901}{100} to get \frac{1200}{901}-\frac{40}{901}i.

$$n=\frac{1200}{901}-\frac{40}{901}i$$

The system is now solved.

$$g=-\frac{1}{10}i n=\frac{1200}{901}-\frac{40}{901}i h=i$$