Solve (lambda^2+1)x+(lambda-1)y-3lambda^2+lambda-4=0

$$( \lambda ^ { 2 } + 1 ) x + ( \lambda - 1 ) y - 3 \lambda ^ { 2 } + \lambda - 4 = 0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{y\lambda -y-3\lambda ^{2}+\lambda -4}{\lambda ^{2}+1}\text{, }&\lambda \neq -i\text{ and }\lambda \neq i\\x\in \mathrm{C}\text{, }&\left(\lambda =i\text{ or }\lambda =-i\right)\text{ and }y=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda ^{2}+1 by x.

$$\lambda ^{2}x+x+\left(\lambda -1\right)y-3\lambda ^{2}+\lambda -4=0$$

Use the distributive property to multiply \lambda -1 by y.

$$\lambda ^{2}x+x+\lambda y-y-3\lambda ^{2}+\lambda -4=0$$

Subtract \lambda y from both sides. Anything subtracted from zero gives its negation.

$$\lambda ^{2}x+x-y-3\lambda ^{2}+\lambda -4=-\lambda y$$

Add y to both sides.

$$\lambda ^{2}x+x-3\lambda ^{2}+\lambda -4=-\lambda y+y$$

Add 3\lambda ^{2} to both sides.

$$\lambda ^{2}x+x+\lambda -4=-\lambda y+y+3\lambda ^{2}$$

Subtract \lambda from both sides.

$$\lambda ^{2}x+x-4=-\lambda y+y+3\lambda ^{2}-\lambda$$

Add 4 to both sides.

$$\lambda ^{2}x+x=-\lambda y+y+3\lambda ^{2}-\lambda +4$$

Reorder the terms.

$$x\lambda ^{2}+x=-y\lambda +y+3\lambda ^{2}-\lambda +4$$

Combine all terms containing x.

$$\left(\lambda ^{2}+1\right)x=-y\lambda +y+3\lambda ^{2}-\lambda +4$$

The equation is in standard form.

$$\left(\lambda ^{2}+1\right)x=4-\lambda +3\lambda ^{2}+y-y\lambda$$

Divide both sides by \lambda ^{2}+1.

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

Dividing by \lambda ^{2}+1 undoes the multiplication by \lambda ^{2}+1.

$$x=\frac{4-\lambda +3\lambda ^{2}+y-y\lambda }{\lambda ^{2}+1}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=-\frac{x\lambda ^{2}+x-3\lambda ^{2}+\lambda -4}{\lambda -1}\text{, }&\lambda \neq 1\\y\in \mathrm{C}\text{, }&x=3\text{ and }\lambda =1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda ^{2}+1 by x.

$$\lambda ^{2}x+x+\left(\lambda -1\right)y-3\lambda ^{2}+\lambda -4=0$$

Use the distributive property to multiply \lambda -1 by y.

$$\lambda ^{2}x+x+\lambda y-y-3\lambda ^{2}+\lambda -4=0$$

Subtract \lambda ^{2}x from both sides. Anything subtracted from zero gives its negation.

$$x+\lambda y-y-3\lambda ^{2}+\lambda -4=-\lambda ^{2}x$$

Subtract x from both sides.

$$\lambda y-y-3\lambda ^{2}+\lambda -4=-\lambda ^{2}x-x$$

Add 3\lambda ^{2} to both sides.

$$\lambda y-y+\lambda -4=-\lambda ^{2}x-x+3\lambda ^{2}$$

Subtract \lambda from both sides.

$$\lambda y-y-4=-\lambda ^{2}x-x+3\lambda ^{2}-\lambda$$

Add 4 to both sides.

$$\lambda y-y=-\lambda ^{2}x-x+3\lambda ^{2}-\lambda +4$$

Reorder the terms.

$$y\lambda -y=-x\lambda ^{2}-x+3\lambda ^{2}-\lambda +4$$

Combine all terms containing y.

$$\left(\lambda -1\right)y=-x\lambda ^{2}-x+3\lambda ^{2}-\lambda +4$$

The equation is in standard form.

$$\left(\lambda -1\right)y=4-\lambda +3\lambda ^{2}-x-x\lambda ^{2}$$

Divide both sides by \lambda -1.

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

Dividing by \lambda -1 undoes the multiplication by \lambda -1.

$$y=\frac{4-\lambda +3\lambda ^{2}-x-x\lambda ^{2}}{\lambda -1}$$

Solve for x

$$x=-\frac{y\lambda -y-3\lambda ^{2}+\lambda -4}{\lambda ^{2}+1}$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda ^{2}+1 by x.

$$\lambda ^{2}x+x+\left(\lambda -1\right)y-3\lambda ^{2}+\lambda -4=0$$

Use the distributive property to multiply \lambda -1 by y.

$$\lambda ^{2}x+x+\lambda y-y-3\lambda ^{2}+\lambda -4=0$$

Subtract \lambda y from both sides. Anything subtracted from zero gives its negation.

$$\lambda ^{2}x+x-y-3\lambda ^{2}+\lambda -4=-\lambda y$$

Add y to both sides.

$$\lambda ^{2}x+x-3\lambda ^{2}+\lambda -4=-\lambda y+y$$

Add 3\lambda ^{2} to both sides.

$$\lambda ^{2}x+x+\lambda -4=-\lambda y+y+3\lambda ^{2}$$

Subtract \lambda from both sides.

$$\lambda ^{2}x+x-4=-\lambda y+y+3\lambda ^{2}-\lambda$$

Add 4 to both sides.

$$\lambda ^{2}x+x=-\lambda y+y+3\lambda ^{2}-\lambda +4$$

Reorder the terms.

$$x\lambda ^{2}+x=-y\lambda +y+3\lambda ^{2}-\lambda +4$$

Combine all terms containing x.

$$\left(\lambda ^{2}+1\right)x=-y\lambda +y+3\lambda ^{2}-\lambda +4$$

The equation is in standard form.

$$\left(\lambda ^{2}+1\right)x=4-\lambda +3\lambda ^{2}+y-y\lambda$$

Divide both sides by \lambda ^{2}+1.

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

Dividing by \lambda ^{2}+1 undoes the multiplication by \lambda ^{2}+1.

$$x=\frac{4-\lambda +3\lambda ^{2}+y-y\lambda }{\lambda ^{2}+1}$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{x\lambda ^{2}+x-3\lambda ^{2}+\lambda -4}{\lambda -1}\text{, }&\lambda \neq 1\\y\in \mathrm{R}\text{, }&x=3\text{ and }\lambda =1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda ^{2}+1 by x.

$$\lambda ^{2}x+x+\left(\lambda -1\right)y-3\lambda ^{2}+\lambda -4=0$$

Use the distributive property to multiply \lambda -1 by y.

$$\lambda ^{2}x+x+\lambda y-y-3\lambda ^{2}+\lambda -4=0$$

Subtract \lambda ^{2}x from both sides. Anything subtracted from zero gives its negation.

$$x+\lambda y-y-3\lambda ^{2}+\lambda -4=-\lambda ^{2}x$$

Subtract x from both sides.

$$\lambda y-y-3\lambda ^{2}+\lambda -4=-\lambda ^{2}x-x$$

Add 3\lambda ^{2} to both sides.

$$\lambda y-y+\lambda -4=-\lambda ^{2}x-x+3\lambda ^{2}$$

Subtract \lambda from both sides.

$$\lambda y-y-4=-\lambda ^{2}x-x+3\lambda ^{2}-\lambda$$

Add 4 to both sides.

$$\lambda y-y=-\lambda ^{2}x-x+3\lambda ^{2}-\lambda +4$$

Reorder the terms.

$$y\lambda -y=-x\lambda ^{2}-x+3\lambda ^{2}-\lambda +4$$

Combine all terms containing y.

$$\left(\lambda -1\right)y=-x\lambda ^{2}-x+3\lambda ^{2}-\lambda +4$$

The equation is in standard form.

$$\left(\lambda -1\right)y=4-\lambda +3\lambda ^{2}-x-x\lambda ^{2}$$

Divide both sides by \lambda -1.

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

Dividing by \lambda -1 undoes the multiplication by \lambda -1.

$$y=\frac{4-\lambda +3\lambda ^{2}-x-x\lambda ^{2}}{\lambda -1}$$