Algebra
Suppose Clara buys $b$ books and $m$ magazines. Then we know that, working in pence:
$340b+160m=2300$
Both sides of this equation can be divided by $20$, to give:
$17b+8m=115$
At this point, it is worth remembering that $m$ and $b$ must both be non-negative integers.
Since $8m$ is even and $115$ is odd, $17b$ must be odd, which in turn means that $b$ must be odd. Also, as $17 \times 7 = 119 > 115$, $b$ must be smaller than $7$. This means the possible values for be are $1$, $3$ and $5$.
If $b=1$, $17+8m=115$, so $8m = 98$, but $98$ is not divisible by $8$. So, $b \neq 1$.
If $b=3$, $51+8m=115$, so $8m = 64$ and $m=8$. This gives one solution.
If $b=5$, $85+8m=115$, so $8m = 30$, but $30$ is not divisible by $8$. So, $b \neq 5$.
Therefore the only solution is that Clara bought $3$ books and $8$ magazines.