Word problem - set up linear equation Four friends, Andrew, Bob, Chris and David, all have different heights. The sum of their heights is 670 cm.
Andrew is 8cm taller than Chris and Bob is 4cm shorter than David.
The sum of the heights of the tallest and shortest of the friends is 2cm more than the sum of the heights of the other two.
Find the height of each friend.
 A: Let the heights of andrew bob chris and david be $a,b,c,d$. Then
$$a+b+c+d=670$$
$$a=c+8$$
$$d=b+4$$
This tells us that either $a$ or $d$ is the tallest height and $c$ or $b$ the shortest. So there are in total $4$ options try each of them and see if all the constraints are satisfied. For example if andrew is tallest and chris is shortest.
We have
$$a+c=2+b+d$$
This gives c=164,a=172,b=165,d=169. This fits all the constraints. This is one possible solution. Try similar other 3 possibilities.
A: Let us dnote there heights by a,b,c,d resp. then equation are 
$$a+b+c+d=670$$ $$c+8=a$$ $$b+4=d$$ they together gives $$2c+2b+12=670 \implies c+b=329$$ 
Now  Andrew is 8cm taller than Chris and Bob is 4cm shorter than David says that chris is not tallest and neither is bob, which says andrew or david is tallest and shortest is either bob or chris. 
So one more equation will be either $$a+c=b+d+2$$ or $$d+c=a+b+2$$ or $$a+b=c+d+2$$ or $$d+b=a+c+2$$
solve all cases
A: You are given that $$\begin{cases}A=C+8\\B=D-4\\A+B+C+D=670\\A\neq B\neq C\neq D\end{cases}$$ Substituting the first two in the third you obtain $$C+D=333$$ therefore
$$\begin{cases}A=C+8\\B=329-C\\D=333-C\\A\neq B\neq C\neq D\end{cases}$$
Now compare the sums by pairs to check which pairs satisfy the condition in the second to last sentence (i.e. that sums differ by 2): $$A+B=337 \qquad \text{ vs } \qquad C+D=333 \qquad 337-333=4\neq2\text{ no }$$ $$A+D=341 \qquad \text{ vs } \qquad B+C=329 \qquad 341-329=12\neq2\text{ no }$$ So, since $A>C$ the only possibility is that $A$ is the tallest and $D$ the shortest  $$A+C=2C+8 \qquad \text{ vs } \qquad B+D=662-2C \qquad \text{ yes }$$ which gives $$2C+8=662-2C+2 \implies 4C=656 \implies C=164$$
A: Hint:  let the heights be $a,b,c,d$.  The second and third sentences give you three equations-can you write them?  From the fourth sentence, we know the tallest is either Andrew or David and the shortest is either Bob or Chris.  That gives you four possibilities for the fourth equation.  All the equations are linear, use substitution.
