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

$$y = t ( x ^ { 2 } - 3 x + 2 ) + ( 1 - t ) ( - 2 x + 4 )$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=-\frac{2x+y-4}{\left(2-x\right)\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 2\\t\in \mathrm{C}\text{, }&\left(y=0\text{ and }x=2\right)\text{ or }\left(y=6\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply t by x^{2}-3x+2.

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

Use the distributive property to multiply 1-t by -2x+4.

$$y=tx^{2}-3tx+2t-2x+4+2tx-4t$$

Combine -3tx and 2tx to get -tx.

$$y=tx^{2}-tx+2t-2x+4-4t$$

Combine 2t and -4t to get -2t.

$$y=tx^{2}-tx-2t-2x+4$$

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

$$tx^{2}-tx-2t-2x+4=y$$

Add 2x to both sides.

$$tx^{2}-tx-2t+4=y+2x$$

Subtract 4 from both sides.

$$tx^{2}-tx-2t=y+2x-4$$

Combine all terms containing t.

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

The equation is in standard form.

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

Divide both sides by x^{2}-x-2.

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

Dividing by x^{2}-x-2 undoes the multiplication by x^{2}-x-2.

$$t=\frac{2x+y-4}{x^{2}-x-2}$$

Divide 2x+y-4 by x^{2}-x-2.

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

Solve for t

$$\left\{\begin{matrix}t=-\frac{2x+y-4}{\left(2-x\right)\left(x+1\right)}\text{, }&x\neq -1\text{ and }x\neq 2\\t\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=2\right)\text{ or }\left(y=6\text{ and }x=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply t by x^{2}-3x+2.

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

Use the distributive property to multiply 1-t by -2x+4.

$$y=tx^{2}-3tx+2t-2x+4+2tx-4t$$

Combine -3tx and 2tx to get -tx.

$$y=tx^{2}-tx+2t-2x+4-4t$$

Combine 2t and -4t to get -2t.

$$y=tx^{2}-tx-2t-2x+4$$

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

$$tx^{2}-tx-2t-2x+4=y$$

Add 2x to both sides.

$$tx^{2}-tx-2t+4=y+2x$$

Subtract 4 from both sides.

$$tx^{2}-tx-2t=y+2x-4$$

Combine all terms containing t.

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

The equation is in standard form.

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

Divide both sides by x^{2}-x-2.

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

Dividing by x^{2}-x-2 undoes the multiplication by x^{2}-x-2.

$$t=\frac{2x+y-4}{x^{2}-x-2}$$

Divide 2x+y-4 by x^{2}-x-2.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{4ty+9t^{2}-12t+4}-t-2}{2t}\text{; }x=\frac{\sqrt{4ty+9t^{2}-12t+4}+t+2}{2t}\text{, }&t\neq 0\\x=\frac{4-y}{2}\text{, }&t=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4ty+9t^{2}-12t+4}-t-2}{2t}\text{; }x=\frac{\sqrt{4ty+9t^{2}-12t+4}+t+2}{2t}\text{, }&\left(t>0\text{ or }y\leq -\frac{\left(3t-2\right)^{2}}{4t}\right)\text{ and }\left(y\leq \text{Indeterminate}\text{ or }t\neq 0\right)\text{ and }\left(t<0\text{ or }\left(t\neq 0\text{ and }y\geq -\frac{\left(3t-2\right)^{2}}{4t}\right)\right)\\x=\frac{4-y}{2}\text{, }&t=0\end{matrix}\right.$$