Solve {l}{f{(3)}=5}{{(2f{(x)}-g{(x)})}=4}{h=f}{i=g}{j=g{(3)}}{text{Solvefor}k,l,mtext{where}}{k=h}{l=i}{m=j}

$$\left. \begin{array} { l } { f {(3)} = 5 }\\ { {(2 f {(x)} - g {(x)})} = 4 }\\ { h = f }\\ { i = g }\\ { j = g {(3)} }\\ { \text{Solve for } k,l,m \text{ where} } \\ { k = h }\\ { l = i }\\ { m = j } \end{array} \right.$$

Solve for f, x, g, h, j, k, l, m

$$k = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$l=i$$

$$m=3i$$

Solution Steps

Consider the first equation. Divide both sides by 3.

$$f=\frac{5}{3}$$

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

$$h=\frac{5}{3}$$

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

$$g=i$$

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

$$j=3i$$

Consider the equation (6). Insert the known values of variables into the equation.

$$k=\frac{5}{3}$$

Consider the equation (8). Insert the known values of variables into the equation.

$$m=3i$$

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

$$2\times \frac{5}{3}x-ix=4$$

Multiply 2 and \frac{5}{3} to get \frac{10}{3}.

$$\frac{10}{3}x-ix=4$$

Combine \frac{10}{3}x and -ix to get \left(\frac{10}{3}-i\right)x.

$$\left(\frac{10}{3}-i\right)x=4$$

Divide both sides by \frac{10}{3}-i.

$$x=\frac{4}{\frac{10}{3}-i}$$

Multiply both numerator and denominator of \frac{4}{\frac{10}{3}-i} by the complex conjugate of the denominator, \frac{10}{3}+i.

$$x=\frac{4\left(\frac{10}{3}+i\right)}{\left(\frac{10}{3}-i\right)\left(\frac{10}{3}+i\right)}$$

Do the multiplications in \frac{4\left(\frac{10}{3}+i\right)}{\left(\frac{10}{3}-i\right)\left(\frac{10}{3}+i\right)}.

$$x=\frac{\frac{40}{3}+4i}{\frac{109}{9}}$$

Divide \frac{40}{3}+4i by \frac{109}{9} to get \frac{120}{109}+\frac{36}{109}i.

$$x=\frac{120}{109}+\frac{36}{109}i$$

The system is now solved.

$$f=\frac{5}{3} x=\frac{120}{109}+\frac{36}{109}i g=i h=\frac{5}{3} j=3i k=\frac{5}{3} l=i m=3i$$