Solve {l}{f{(x)}=3x-1}{g=f{(-2)}}{h=g}{text{Solvefor}itext{where}}{i=h}

$$\left. \begin{array} { l } { f {(x)} = 3 x - 1 }\\ { g = f {(-2)} }\\ { h = g }\\ { \text{Solve for } i \text{ where} } \\ { i = h } \end{array} \right.$$

Solve for f, x, g, h

$$x=\frac{12}{37}-\frac{2}{37}i\approx 0.324324324-0.054054054i$$

$$f=-\frac{1}{2}i=-0.5i$$

$$g=i$$

$$h=i$$

Solution Steps

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

$$h=i$$

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

$$i=g$$

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

$$g=i$$

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

$$i=f\left(-2\right)$$

Divide both sides by -2.

$$\frac{i}{-2}=f$$

Divide i by -2 to get -\frac{1}{2}i.

$$-\frac{1}{2}i=f$$

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

$$f=-\frac{1}{2}i$$

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

$$-\frac{1}{2}ix=3x-1$$

Subtract 3x from both sides.

$$-\frac{1}{2}ix-3x=-1$$

Combine -\frac{1}{2}ix and -3x to get \left(-3-\frac{1}{2}i\right)x.

$$\left(-3-\frac{1}{2}i\right)x=-1$$

Divide both sides by -3-\frac{1}{2}i.

$$x=\frac{-1}{-3-\frac{1}{2}i}$$

Multiply both numerator and denominator of \frac{-1}{-3-\frac{1}{2}i} by the complex conjugate of the denominator, -3+\frac{1}{2}i.

$$x=\frac{-\left(-3+\frac{1}{2}i\right)}{\left(-3-\frac{1}{2}i\right)\left(-3+\frac{1}{2}i\right)}$$

Do the multiplications in \frac{-\left(-3+\frac{1}{2}i\right)}{\left(-3-\frac{1}{2}i\right)\left(-3+\frac{1}{2}i\right)}.

$$x=\frac{3-\frac{1}{2}i}{\frac{37}{4}}$$

Divide 3-\frac{1}{2}i by \frac{37}{4} to get \frac{12}{37}-\frac{2}{37}i.

$$x=\frac{12}{37}-\frac{2}{37}i$$

The system is now solved.

$$f=-\frac{1}{2}i x=\frac{12}{37}-\frac{2}{37}i g=i h=i$$