Answer: There are four possible solutions
- a = 3, b = 5, c = 6, d = 2
- a = 4, b = 4, c = 3, d = 4
- a = 4, b = 4, c = 4, d = 3
- a = 5, b = 3, c = 2, d = 6
Step-by-step explanation:
I was unable to manipulate this system to develop solvable equations, so I made a table consisting of 7 rows with the following 5 columns:
- a (values of 1 thru 7)
- b (values of 7 thru 1)
- a·b (multiplying column 1 by column 2)
- c + d (subtracting column 3 from 23)
- cd=12 (two addends from column 4 that create a product of 12)
Table 1
[tex]\begin{array}{c|c|c|c|c}{a&b&ab&c+d&cd\\1&7&7&16&\text{null}\\2&6&12&11&\text{null}\\3&5&15&8&2,6\\4&4&16&7&3,4\\5&3&15&8&6,2\\6&2&12&11&\text{null}\\7&1&7&16&\text{null}\\\end{array}[/tex]
I used the valid solutions (rows 3 thru 5) to determine which combinations satisfied ab + cd = 28.
Table 2
[tex]\begin{array}{c|c|c|c|c}a&b&c&d&ab+cd\\3&5&2&6&6+11=17\\3&5&6&2&18+10=28^*\\4&4&3&4&12+16=28^*\\4&4&4&3&16+12=28^*\\5&3&2&6&10+18=28^*\\5&3&6&2&30+6=36\\\end{array}[/tex]
* represent the valid solutions
