Solve {l}{f{(1-x/2+x)}=1-4}{g=f{(3)}}{h=g}{i=h}{text{Solvefor}jtext{where}}{j=i}

$$\left. \begin{array} { l } { f {(\frac{1 - x}{2 + x})} = 1 - 4 }\\ { g = f {(3)} }\\ { h = g }\\ { i = h }\\ { \text{Solve for } j \text{ where} } \\ { j = i } \end{array} \right.$$

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.

$$i=f\times 3$$

Divide both sides by 3.

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

Divide i by 3 to get \frac{1}{3}i.

$$\frac{1}{3}i=f$$

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

$$f=\frac{1}{3}i$$

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

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

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$\frac{1}{3}i\left(1-x\right)=x+2+\left(x+2\right)\left(-4\right)$$

Use the distributive property to multiply \frac{1}{3}i by 1-x.

$$\frac{1}{3}i-\frac{1}{3}ix=x+2+\left(x+2\right)\left(-4\right)$$

Use the distributive property to multiply x+2 by -4.

$$\frac{1}{3}i-\frac{1}{3}ix=x+2-4x-8$$

Combine x and -4x to get -3x.

$$\frac{1}{3}i-\frac{1}{3}ix=-3x+2-8$$

Subtract 8 from 2 to get -6.

$$\frac{1}{3}i-\frac{1}{3}ix=-3x-6$$

Add 3x to both sides.

$$\frac{1}{3}i-\frac{1}{3}ix+3x=-6$$

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

$$\frac{1}{3}i+\left(3-\frac{1}{3}i\right)x=-6$$

Subtract \frac{1}{3}i from both sides.

$$\left(3-\frac{1}{3}i\right)x=-6-\frac{1}{3}i$$

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

$$x=\frac{-6-\frac{1}{3}i}{3-\frac{1}{3}i}$$

Multiply both numerator and denominator of \frac{-6-\frac{1}{3}i}{3-\frac{1}{3}i} by the complex conjugate of the denominator, 3+\frac{1}{3}i.

$$x=\frac{\left(-6-\frac{1}{3}i\right)\left(3+\frac{1}{3}i\right)}{\left(3-\frac{1}{3}i\right)\left(3+\frac{1}{3}i\right)}$$

Do the multiplications in \frac{\left(-6-\frac{1}{3}i\right)\left(3+\frac{1}{3}i\right)}{\left(3-\frac{1}{3}i\right)\left(3+\frac{1}{3}i\right)}.

$$x=\frac{-\frac{161}{9}-3i}{\frac{82}{9}}$$

Divide -\frac{161}{9}-3i by \frac{82}{9} to get -\frac{161}{82}-\frac{27}{82}i.

$$x=-\frac{161}{82}-\frac{27}{82}i$$

The system is now solved.

$$f=\frac{1}{3}i x=-\frac{161}{82}-\frac{27}{82}i g=i h=i j=i$$