Here’s a challenge!
In a problem as messy and lengthy as this, it’s vital to check the derived solutions. Here are the original
equations once again, for reference:
− 4 x+ 2 y− 3 z= 5
2 x− 5 y=z− 1
3 x=− 6 y+ 7 z
“So,” you ask, “What’s the challenge here? It’s only arithmetic!” The answer: “It’s a challenge to force our-
selves through the tedium. But it has to be done if we want to be sure our answers are correct.”
Solution
Let’s start with the first equation. Plugging in the values for x,y, and z, and then doing the arithmetic
carefully, we pass through the following steps:
− 4 x+ 2 y− 3 z= 5
− 4 × (−209/223)+ 2 × (−18/223)− 3 × (−105/223)= 5
836/223− 36/223 + 315/223 = 5
(836− 36 + 315)/223 = 5
1,115/223= 5
5 = 5So far, we’re doing okay. Now for the second equation check:
2 x− 5 y=z− 1
2 × (−209/223)− 5 × (−18/223)=−105/223− 1
−418/223+ 90/223 =−105/223− 223/223
(− 418 + 90)/223 = (− 105 − 223)/223
−328/223=−328/223
Two checks are done, and one remains. Here we go:
3 x=− 6 y+ 7 z
3 × (−209/223)=− 6 × (−18/223)+ 7 × (−105/223)
−627/223= 108/223 − 735/223
−627/223= (108 − 735)/223
−627/223=−627/223
Mission accomplished!
Substitute Back 289