https://gyazo.com/9bafc92df7fe8b36d732d9834c04ec54 How do you set up the system
ID: 3138608 • Letter: H
Question
https://gyazo.com/9bafc92df7fe8b36d732d9834c04ec54
How do you set up the system of equations for thhis problem so oyu can set up an augmented matrix?
Sandra borrowed $19,500...
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(I answered the problem: X being the Parents at 6%, Y being the Credit Union at 7%, and Z being the Bank at 5%. Knowing that Sandra borrowed 5 times as much as she did from the Bank, I set Z= $1,000, $2,000, and $3,000; thus appropriately so, X is $5,000, $10,000, and $15,000. Z can't be $4,000 since X would have to = $20,000 which exceeds the $19,500 total. Thus, Y= $19,500 - (X+Z): $13,500, $7,500, and $1,500. From there, use simple interest fromula of I=Prt, where I is the interest paid, P is the amount borrowed, r is the interest rate, and t is time in years, in this case 1. So, ($10,000(.06))+($7,500(.07))+($2,000(.05))= $1,225, the total amount paid in interest, and $10,000 + $7,500 + $2,000 = $19,500. While this did work out since the numbers were pretty, if you could explain it in a better approach where you don't guess and check, that would be great, thanks.)
Explanation / Answer
Let the amounts borrowed by Sandra from her parents, the credit union and the bank be $ x,$y and $ z respectively. Then x+y+z = 19500…(1)
Also x = 5z …(2). Further, the interest for 1 year on the borrowed sum is x*0.06+y*0.07+z*0.05 = 1225. On multiplying both the sides by 100, we get 6x+7y+5z = 122500…(3).
On substituting x = 5z in the 1st and the 3rd equations, we get 5z+y+z = 19500 or, y+6z = 19500…(4) and 6*5z +7y+5z = 122500 or, 7y +35z = 122500 or, on diving both the sides by 7, we get y +5z = 17500…(5).
Now, on subtracting the 5th equation from the 4th equation, we get Y+6z-y-5z = 19500-17500 or,z = 2000. Then, from the 5th equation, we get y + 5*2000 = 17500 or, y +10000 = 17500 so that y = 7500. Further, from the 2nd equation, we get x = 5z = 5*2000 = 10000.
Thus, Sandra borrowed $ 10000 from her parents, $ 7500 from the credit union and $ 2000 from the bank.
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