Solve {l}{f{(x)}=4x+5}{g=f{(5)}}{h=g}{i=h}{text{Solvefor}jtext{where}}{j=i}

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

Solve for f, x, g, h, j

$$j=i$$

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.

$$i=f\times 5$$

Divide both sides by 5.

$$\frac{i}{5}=f$$

Divide i by 5 to get \frac{1}{5}i.

$$\frac{1}{5}i=f$$

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

$$f=\frac{1}{5}i$$

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

$$\frac{1}{5}ix=4x+5$$

Subtract 4x from both sides.

$$\frac{1}{5}ix-4x=5$$

Combine \frac{1}{5}ix and -4x to get \left(-4+\frac{1}{5}i\right)x.

$$\left(-4+\frac{1}{5}i\right)x=5$$

Divide both sides by -4+\frac{1}{5}i.

$$x=\frac{5}{-4+\frac{1}{5}i}$$

Multiply both numerator and denominator of \frac{5}{-4+\frac{1}{5}i} by the complex conjugate of the denominator, -4-\frac{1}{5}i.

$$x=\frac{5\left(-4-\frac{1}{5}i\right)}{\left(-4+\frac{1}{5}i\right)\left(-4-\frac{1}{5}i\right)}$$

Do the multiplications in \frac{5\left(-4-\frac{1}{5}i\right)}{\left(-4+\frac{1}{5}i\right)\left(-4-\frac{1}{5}i\right)}.

$$x=\frac{-20-i}{\frac{401}{25}}$$

Divide -20-i by \frac{401}{25} to get -\frac{500}{401}-\frac{25}{401}i.

$$x=-\frac{500}{401}-\frac{25}{401}i$$

The system is now solved.

$$f=\frac{1}{5}i x=-\frac{500}{401}-\frac{25}{401}i g=i h=i j=i$$