Let X be the amount borrowed from the bank and Y be the amount borrowed from the life insurance.
X + Y = 100,000
as he borrowed $100,000
0.10X + 0.12Y = 10900
10% = 0.10 and 12% = 0.12;
The total interest is $10,900
Solving the first equation for Y and multiplying the second equation by 100 to clear the decimals:
Y = 100,000 - X
10X + 12Y = 1090000
Plugging the first equation into the second:
10X + 12(100,000 - X) = 1090000
10X + 1200000 - 12X = 1090000 -------- distributive property
0 = 12X - 10X + 1090000 - 1200000 --- moves everything to right side, signs change; like to keep coefficient positive
0 = 2X -110000 --- combines like terms
110000 = 2x ---- solves for X
110000/2 = X --- adds 180000 to both sides; then divides by 4
X = 55000
Y = 100000 - 55000 = 45000
CHECK: 55000 x 10% + 45000 x 12% = 55000 x 0.10 + 45000 x 0.12
= 5500 + 5400 = 10,900
He borrowed 55000 from the bank and 45000 from the insurance.
