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March 26th, 2016, 05:26 PM  #1 
Newbie Joined: Mar 2016 From: Canada Posts: 24 Thanks: 0  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. 
March 26th, 2016, 06:13 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,002 Thanks: 1587 
Each equation represents a plane in 3D ... $\begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & 3\\ 4 & 5 & 1 \end{bmatrix} \cdot \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}= \begin{bmatrix} 6\\ 5\\ 3\\ \end{bmatrix}$ 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? 
March 27th, 2016, 02:41 PM  #3 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 660 Thanks: 87 
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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