# Thread: Parallel and intersective lines solve for the intersections

1. ## Parallel and intersective lines solve for the intersections

I'm stuck on this question, would appreciate any help trying to figure this out:

Q)Two parallel lines $\displaystyle y = 2x + 2$ and $\displaystyle y = 2x - 2$ are crossed by a third line. Determine the coefficients of the third line $\displaystyle y = ax + b$ such that the x-coordinate of the intersection of the two lines $\displaystyle y = ax + b$ and $\displaystyle y = 2x - 2$ is twice as much as that of the intersection of the two lines $\displaystyle y = ax + b$ and $\displaystyle y = 2x + 2$; and the y-coordinate of the intersection of the two lines $\displaystyle y = ax + b$ and $\displaystyle y = 2x + 2$ is twice as much as that of the intersection of the two lines $\displaystyle y = ax + b$ and $\displaystyle y = 2x - 2$.
Calculate the coordinates of the two intersections.

I would appreciate any help on this.

2. ## Re: Parallel and intersective lines solve for the intersections

Solving $\displaystyle \begin{Bmatrix} y=ax+b\\y=2x+2\end{matrix}$ you'll obtain $\displaystyle x_1=\frac{2-b}{a-2},\;y_1=\frac{2(a-b)}{a-2}$ .

Solving $\displaystyle \begin{Bmatrix} y=ax+b\\y=2x-2\end{matrix}$ you'll obtain $\displaystyle x_2=\frac{b+2}{2-a},\;y_2=\frac{2(a+b)}{2-a}$ .

Now, using $\displaystyle x_2=2x_1,\;y_1=2y_2$ we get a $\displaystyle 2\times 2$ linear system. Find $\displaystyle a$ and $\displaystyle b$ .

Thank you.