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

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

Solve for f, x, g, h

$$x=\frac{24}{37}+\frac{4}{37}i\approx 0.648648649+0.108108108i$$

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

$$g=i$$

$$h=i$$

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.

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

Add 3x to both sides.

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

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

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

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

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

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

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

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

$$x=\frac{6+i}{\frac{37}{4}}$$

Divide 6+i by \frac{37}{4} to get \frac{24}{37}+\frac{4}{37}i.

$$x=\frac{24}{37}+\frac{4}{37}i$$

The system is now solved.

$$f=-\frac{1}{2}i x=\frac{24}{37}+\frac{4}{37}i g=i h=i$$