How to write a bird population word problem as a matrix

Key concept: This is a clean translation problem: three groups, three conditions, one system. The trick is to name the variables in a way that makes the relationships easy to read.

Step by step

Let

• $s$ = number of sapsuckers
• $n$ = number of nutcrackers
• $h$ = number of honeyguzzlers

We are told:

1. There are 1080 birds total: $s+n+h=1080$
2. There are twice as many nutcrackers as honeyguzzlers: $n=2h$ which we can rewrite as $n-2h=0$
3. There are 120 more sapsuckers than nutcrackers: $s=n+120$ which becomes $s-n=120$

System in standard form

s+n+h=1080\\ s-n=120\\ n-2h=0 \end{cases}$$ ### Augmented matrix form Using variable order $s,n,h$: $$\begin{bmatrix} 1 & 1 & 1 & | & 1080\\ 1 & -1 & 0 & | & 120\\ 0 & 1 & -2 & | & 0 \end{bmatrix}$$ ### Pitfall alert A frequent mistake is writing the second relationship as $h=2n$ instead of $n=2h$. Read the phrase carefully: ‘twice as many nutcrackers as honeyguzzlers’ means nutcrackers are the larger group. Also keep the variable order consistent across every row of the matrix. ### Try different conditions If the problem asked for the actual number of each bird, you could continue by row-reducing the matrix or using substitution. With the current wording, though, only the system and augmented matrix are required. If a different species relationship were given, the matrix coefficients would change but the setup method would stay the same. ### Further reading augmented matrix, system of equations, word problem