Solve {l}{g{(x)}=3x-3}{h=g{(-6)}}{text{Solvefor}itext{where}}{i=h}

$$\left. \begin{array} { l } { g {(x)} = 3 x - 3 }\\ { h = g {(-6)} }\\ { \text{Solve for } i \text{ where} } \\ { i = h } \end{array} \right.$$

Solve for g, x, h

$$x=\frac{324}{325}-\frac{18}{325}i\approx 0.996923077-0.055384615i$$

$$g=-\frac{1}{6}i\approx -0.166666667i$$

$$h=i$$

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.

$$i=g\left(-6\right)$$

Divide both sides by -6.

$$\frac{i}{-6}=g$$

Divide i by -6 to get -\frac{1}{6}i.

$$-\frac{1}{6}i=g$$

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

$$g=-\frac{1}{6}i$$

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

$$-\frac{1}{6}ix=3x-3$$

Subtract 3x from both sides.

$$-\frac{1}{6}ix-3x=-3$$

Combine -\frac{1}{6}ix and -3x to get \left(-3-\frac{1}{6}i\right)x.

$$\left(-3-\frac{1}{6}i\right)x=-3$$

Divide both sides by -3-\frac{1}{6}i.

$$x=\frac{-3}{-3-\frac{1}{6}i}$$

Multiply both numerator and denominator of \frac{-3}{-3-\frac{1}{6}i} by the complex conjugate of the denominator, -3+\frac{1}{6}i.

$$x=\frac{-3\left(-3+\frac{1}{6}i\right)}{\left(-3-\frac{1}{6}i\right)\left(-3+\frac{1}{6}i\right)}$$

Do the multiplications in \frac{-3\left(-3+\frac{1}{6}i\right)}{\left(-3-\frac{1}{6}i\right)\left(-3+\frac{1}{6}i\right)}.

$$x=\frac{9-\frac{1}{2}i}{\frac{325}{36}}$$

Divide 9-\frac{1}{2}i by \frac{325}{36} to get \frac{324}{325}-\frac{18}{325}i.

$$x=\frac{324}{325}-\frac{18}{325}i$$

The system is now solved.

$$g=-\frac{1}{6}i x=\frac{324}{325}-\frac{18}{325}i h=i$$