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Geometrical InterpretationSolve 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? |

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? |

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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