Solve {l}{f{(x)}=-4x-4}{g=f{(-1/5)}}{h=g}{text{Solvefor}itext{where}}{i=h}

$$\left. \begin{array} { l } { f {(x)} = -4 x - 4 }\\ { g = f {(-\frac{1}{5})} }\\ { h = g }\\ { \text{Solve for } i \text{ where} } \\ { i = h } \end{array} \right.$$

Solve for f, x, g, h

$$x=-\frac{16}{41}-\frac{20}{41}i\approx -0.390243902-0.487804878i$$

$$f=-5i$$

$$g=i$$

$$h=i$$

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.

$$i=f\left(-\frac{1}{5}\right)$$

Multiply both sides by -5, the reciprocal of -\frac{1}{5}.

$$-5i=f$$

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

$$f=-5i$$

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

$$-5ix=-4x-4$$

Add 4x to both sides.

$$-5ix+4x=-4$$

Combine -5ix and 4x to get \left(4-5i\right)x.

$$\left(4-5i\right)x=-4$$

Divide both sides by 4-5i.

$$x=\frac{-4}{4-5i}$$

Multiply both numerator and denominator of \frac{-4}{4-5i} by the complex conjugate of the denominator, 4+5i.

$$x=\frac{-4\left(4+5i\right)}{\left(4-5i\right)\left(4+5i\right)}$$

Do the multiplications in \frac{-4\left(4+5i\right)}{\left(4-5i\right)\left(4+5i\right)}.

$$x=\frac{-16-20i}{41}$$

Divide -16-20i by 41 to get -\frac{16}{41}-\frac{20}{41}i.

$$x=-\frac{16}{41}-\frac{20}{41}i$$

The system is now solved.

$$f=-5i x=-\frac{16}{41}-\frac{20}{41}i g=i h=i$$