How to find the coordinates of $A$ and of $B$?

Question

The point $A$ lies on the line $y=2x$ and the point $B$ lies on the line $y=x+3$. The coordinates of the midpoint of $AB$ are (2,6). What are the coordinates of $A$ and of $B$?

Answer 1

Since $A$ lies of $y=2x$, the coordinates of $A$ are $(a,2a)$. Similarly the coordinates of $B$ are $(b,b+3)$.

Using these coordinates in terms of $a,b$ to calculate the midpoint leads to the following system of equations.

$$a+b=4$$ $$2a+b=9$$

$(a,b)=(5,-1)$

$A(5,10)$ and $B(-1,2)$

Answer 2

If $\;A=(x,y)\;,\;\;B=(a,b)\;$ , then

$$(2,6)=\left(\frac{x+a}2\,,\,\frac{y+b}2\right)$$

But you also know that $\;A\;$ is on $\;y=2x\;$ , so $\;A=(x,y)=(x,2x)\;$ , and something similar for $\;B\;$ on the other line .

Fill in details and finish the solution