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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: 2,838 Thanks: 1481 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? March 27th, 2016, 02:41 PM #3 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 637 Thanks: 85 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. Tags geometrical, interpretation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mishin05 Calculus 0 December 18th, 2012 12:33 PM happy21 Applied Math 0 March 16th, 2012 12:10 AM mrhesham Algebra 3 December 5th, 2010 11:51 AM kahalla Advanced Statistics 0 March 9th, 2010 01:24 AM grappler Algebra 1 June 11th, 2009 08:22 AM

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