Gaussian Elimination/ Gauss-Jordan elimination

Question

Andre borrowed $20,000 to buy a truck for his business. He borrowed from his parents who charge him 2% simple interest. He borrowed from a credit union that charges 4% simple interest, and he borrowed from a bank that charges 5% simple interest. He borrowed 5 times as much from his parents as from the bank, and the amount of interest he paid at the end of year 1 was $620. How much did he borrow from each source?

Answer 1

X+Y+Z = 20000

0.02X + 0.04Y +0.05Z = 620

X=5z

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substitutes the 3rd equation into the first two equations

5z+y + z = 20000

0.02(5z) + 0.04y + 0.05z = 620

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simplifying and combining like terms...

y + 6z = 20000

0.04y + 0.15z = 620

the first of these is y = 20000-6z

0.04(20000-6z) + 0.15z = 620

800 - 0.24z + 0.15z = 620

800 - 0.09z = 620

-0.09z = -180

9z = 18000

z = 2000

since y + 6z=20000 --> y = 8000

then z=10000

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Here is the Gaussian Elimination

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1 1 1 | 20000

0.02 0.04 0.05 | 620

1 0 -5 | 0

-row3+row1; row2*100

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0 1 6 | 20000

2 4 5 | 62000

1 0 -5 | 0

-2*row3+row2

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0 1 6 | 20000

0 4 15 | 62000

1 0 -5 | 0

-4*row1+row2

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0 1 6 | 20000

0 0 -9 | -18000

1 0 -5 | 0

z = 2000

y + 6z = 20000 per equation #1

y + 6(2000) = 20000

y + 12000 = 20000

y = 8000

x=10000