1. ## Graphs/equations/point of intersection

The solutions of the equation 2x + 2/x = 7 are the x-coordinates of the points of intersection of the graph of y = x + 2/x and a straight line L
Find the equation of L.

2. First find x in the first equation

$\displaystyle 2x+\frac{2}{x}=7$

$\displaystyle x+\frac{1}{x}=\frac{7}{2}$

$\displaystyle \frac{x^2+1}{x}=\frac{7}{2}$

$\displaystyle x^2+1=\frac{7}{2}x$

$\displaystyle \frac{7}{2}x-\frac{2x^2}{2}=1$

$\displaystyle \frac{7x-2x^2}{2}=1$

$\displaystyle 7x-2x^2=2$

$\displaystyle x^2-\frac{7}{2}x=-1$

$\displaystyle x^2-\frac{7}{2}x+\frac{49}{16}=1+\frac{49}{16}$

$\displaystyle (x-\frac{7}{4})^2=1+\frac{49}{16}$

$\displaystyle x-\frac{7}{4}=\pm{\sqrt{1+\frac{49}{16}}}$

$\displaystyle x=\pm{\sqrt{1+\frac{49}{16}}}+\frac{7}{4}$

So this particular value of x lies on the graph of the equation $\displaystyle y=x+\frac{2}{x}$ and is also a point on the line $\displaystyle L$.

Can you take it from here?

I'll give you a hint.

Plug the values $\displaystyle +{\sqrt{1+\frac{49}{16}}}+\frac{7}{4}$ and $\displaystyle -{\sqrt{1+\frac{49}{16}}}+\frac{7}{4}$ into the equation $\displaystyle y=x+\frac{2}{x}$. They are two points on L.

3. 2x +2/x = 7
subtract x from both sides
x + 2/x = 7 - x

so you need to draw y = x + 2/x and the straight line y = 7 - x

It's as simple as that!!

4. yes