[Algebra]
Problem: The table below gives the final standings for seven of the teams in the English Cricket League in 1998. At the end of the year, each team had played 17 matches and had obtained the total number of points shown in the last column. Each win $W$, each draw $D$, each bonus bowling point $A$, and each bonus batting point $B$ received $w, d, a$ and $b$ points respectively, where $w, d, a$ and $b$ are positive integers. No points are given for a loss. Determine the values of $w, d, a$ and $b$ if total points awarded are given by the formula: Points $=w \times W+d \times D+a \times A+b \times B$.



Final Standings

|  | $W$ | Losses | $D$ | $A$ | $B$ | Points |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Sussex | 6 | 7 | 4 | 30 | 63 | 201 |
| Warks | 6 | 8 | 3 | 35 | 60 | 200 |
| Som | 6 | 7 | 4 | 30 | 54 | 192 |
| Derbys | 6 | 7 | 4 | 28 | 55 | 191 |
| Kent | 5 | 5 | 7 | 18 | 59 | 178 |
| Worcs | 4 | 6 | 7 | 32 | 59 | 176 |
| Glam | 4 | 6 | 7 | 36 | 55 | 176 |

Solution: There are a variety of ways to find the unknowns.

The most efficient way is to choose equations that have like coefficients. Here is one way to solve the problem using this method.

For Sussex: $\quad 6 w+4 d+30 a+63 b=201$

For Som: $\quad 6 w+4 d+30 a+54 b=192$

Subtracting, $\quad 9 b=9 b=1$

If $b=1$

For Derbys: $\quad 6 w+4 d+28 a+55=191$ 

$$
6 w+4 d+28 a=136 \tag{1}
$$

For Sussex: $\quad 6 w+4 d+30 a+63=201$

$$
6 w+4 d+30 a=138 \tag{2}
$$

Subtracting, (2) - (1)

$$
2 a=2
$$

$$
a=1 \text {. }
$$

We can now calculate $d$ and $w$ by substituting $a=1, b=1$ into a pair of equations.

An efficient way of doing this is by substituting $a=1, b=1$ into Som and Worcs.

For Som: $\quad 6 w+4 d+84=192$

$$
6 w+4 d=108 \tag{3}
$$

For Worcs: $\quad 6 w+3 d+85=200$

$$
6 w+3 d=105 \tag{4}
$$

Subtracting, (3) - (4) $\quad d=3$.

Substituting $d=3$ in either (3) or (4), $6 w+4(3)=108$ (substituting in (3))

$$
\begin{aligned}
6 w & =96 \\
w & =16 .
\end{aligned}
$$

Therefore $w=16, d=3, a=b=1$.

Final answer: 16,3,1,1