Let:
• l, be the number of letters recieved
,
• m, be the number of magazines recieved
,
• b, be the number of bills recieved
,
• a, be the number of adds recieved
Since we know that the Moore family received 35 pieces of mail in total, we'll have that:
![l+m+b+a=35](https://tex.z-dn.net/?f=l%2Bm%2Bb%2Ba%3D35)
We'll call this equation equation 1.
Now, we know that they received three more magazines than bills. This way,
![m=b+3](https://tex.z-dn.net/?f=m%3Db%2B3)
We'll call this equation equation 2.
We also know that they recieved five more ads than magazines. This way,
![a=m+5](https://tex.z-dn.net/?f=a%3Dm%2B5)
We'll call this equation equation 3.
For our last eqaution, we know that they recieved the same number of letters as bills. This way,
![l=b](https://tex.z-dn.net/?f=l%3Db)
We'll call this equation equation 4.
First, we'll substitute equation 4 into equation 1:
![\begin{gathered} l+m+b+a=35\rightarrow b+m+b+a=35 \\ \Rightarrow m+a+2b=35 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20l%2Bm%2Bb%2Ba%3D35%5Crightarrow%20b%2Bm%2Bb%2Ba%3D35%20%5C%5C%20%5CRightarrow%20m%2Ba%2B2b%3D35%20%5Cend%7Bgathered%7D)
Now, we'll substitute equation 3:
![\begin{gathered} m+a+2b=35\rightarrow m+m+5+2b=35 \\ \rightarrow2m+2b=30 \\ \Rightarrow m+b=15 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20m%2Ba%2B2b%3D35%5Crightarrow%20m%2Bm%2B5%2B2b%3D35%20%5C%5C%20%5Crightarrow2m%2B2b%3D30%20%5C%5C%20%5CRightarrow%20m%2Bb%3D15%20%5Cend%7Bgathered%7D)
Lastly, we substitute equation 2:
![\begin{gathered} m+b=15\rightarrow b+3+b=15 \\ \rightarrow2b+3=15\rightarrow2b=12\rightarrow b=\frac{12}{6} \\ \\ \Rightarrow b=2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20m%2Bb%3D15%5Crightarrow%20b%2B3%2Bb%3D15%20%5C%5C%20%5Crightarrow2b%2B3%3D15%5Crightarrow2b%3D12%5Crightarrow%20b%3D%5Cfrac%7B12%7D%7B6%7D%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20b%3D2%20%5Cend%7Bgathered%7D)
From equation 4, we'll have that:
![l=b\rightarrow l=2](https://tex.z-dn.net/?f=l%3Db%5Crightarrow%20l%3D2)
This way, we can conclude that they recieved 2 letters.