Solve the simultaneous equations: x+y=5 and x2+y2=13. (b) Find xy

From $(1)$, we have $y = 5-x$.

So, $x=2$ or $x=3$.

If $x=2$, then $y = 5-2 = 3$.
If $x=3$, then $y = 5-3 = 2$.

The solutions are $(x,y) = (2,3)$ and $(x,y) = (3,2)$.
The solutions are $\boxed{x=2, y=3 \text{ or } x=3, y=2}$.

13. (b) Find $xy$

If $x=2$ and $y=3$, then $xy = 2 \times 3 = 6$.
If $x=3$ and $y=2$, then $xy = 3 \times 2 = 6$.

In both cases, $xy$ is the same.

The value of $xy$ is $\boxed{6}$.

14. (a) Find the equation of a line passing through $(2,3)$ and $(4,7)$

The formula for the gradient is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Using points $(x_1, y_1) = (2,3)$ and $(x_2, y_2) = (4,7)$:

Using the point $(2,3)$ and $m=2$:

The equation of the line is $\boxed{y = 2x - 1}$.

14. (b) Find the midpoint of the line

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M$ is given by $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

The midpoint of the line is $\boxed{(3,5)}$.