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March 26th, 2016, 05:26 PM   #1
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Geometrical Interpretation

Solve the following system of equations and give a geometrical interpretation of the result.
x + y + z = 6
2x + y − 3z = -5
4x − 5y + z = −3


What does this question mean by "geometrical interpretation"? And what should I be solving for?

Last edited by skipjack; March 26th, 2016 at 06:45 PM.
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March 26th, 2016, 06:13 PM   #2
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Each equation represents a plane in 3-D ...

$\begin{bmatrix}
1 & 1 & 1\\
2 & 1 & -3\\
4 & -5 & 1
\end{bmatrix} \cdot
\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix}=
\begin{bmatrix}
6\\
-5\\
-3\\
\end{bmatrix}$

solving for $\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix}$ with a calculator because I'm too lazy to perform the row reduction drill (and I've had a couple of beers) ...

$\begin{bmatrix}
x\\
y\\
z\\
\end{bmatrix}=
\begin{bmatrix}
1\\
2\\
3\\
\end{bmatrix}$

think about how three planes could intersect (or not) ... what geometrical idea does that final variable matrix and what it equals represent?
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March 27th, 2016, 02:41 PM   #3
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I don't remember how to work with matrices, which I never liked. I solved for x, y, and z in 18 combined steps (21 if you count the original equations). OE means Original Equation:

1. OE1*4: 4x + 4y + 4z = 24
2. OE2*2: 4x + 2y - 6z = -10
3. Step 1 - Step 2: 2y + 10z = 34
4. Step 2 - Step 3: 4x - 16z = -44
5. Express Step 4 in terms of x: 4x = 16z - 44
6. Substitute Step 5 into OE3: 16z - 44 - 5y + z = -3
7. Combine like terms and write y before z before a constant: -5y + 17z - 44 = -3
8. Add 44 to both sides: -5y + 17z = 41
9. Step 3 multiplied by 5: 10y + 50z = 170
10. Step 8 multiplied by 2: -10y + 34z = 82
11. Step 9 + Step 10: 84z = 252
12. Divide by 84 on both sides: z = 3
13. Substitute z = 3 into Step 5: 4x = 48 - 44
14. Equation 13 divided by 4: x = 12 - 11
15. Combine like terms: x = 1
16. Substitute x = 1 and z = 3 into OE3: 1 + y + 3 = 6
17. Combine like terms and write y before the constants: y + 4 = 6
18. Subtract 4 from both sides: y = 2

I'm not saying that's short or simple to follow, but it works.
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