Eliminate one of these variables to decide whether you have a tent or a book

1. Eliminate one of these variables to decide whether you have a tent or a book This is a tent!

2. Recap the steps: You have 3 equations, what type of solution are they? If it is not unique, complete the following: • Check if they are parallel - Make the coefficients of one of the variables the same. If everything else is the same, except the constant, then they are parallel. You have magic carpets (inconsistent and no solutions) • If two of them are parallel, but the third isn’t, then you have ninja styles (inconsistent and no solutions) • If they are not parallel, you either have tents or books. So they either intersect along a line, or the intersection of 2 planes is parallel to the third.

3. Recap the steps: You have 3 equations, what type of solution are they? If it is not unique, complete the following: 4) Eliminate one of the variables, so that you only get two equations with two variables. These two equations are the lines that two of the planes intersect along. • If these equations are the same, then you have a book – they intercept along the spine of the book (dependent and many solutions). • If these two equations are parallel – the same except for the constant, then you have a tent (inconsistent and no solutions)

4. Summary Inconsistent – no solutions Magic carpets Ninja styles Tent Dependent – many solutions Book Same planes

6. practice Investigate the nature of the solutions (roots) for the following simultaneous equations. • 8x + 8y + 3z = 8 6x + 5y + 3z = 4 4x + 2y + 3z = 0 • 2x – 4y + 3z = 6 12x – 24y + 18z = 10 4x – 4y + 6z = 12 This is an example of a system of non-parallel planes that intersect at a line (a book). The system is dependent and has many solutions (any point on the line of intersection). This is an example of a system of two parallel planes cut by a third plane (ninja). The system is inconsistent and has no solutions.

7. Some practice • Complete the questions on the worksheet. • Then, in your homework books, do page 30 (just questions 53 and 54) and decide which type of solution you have got. Also do page 27 (questions 47-49). • Remember: check to see whether they are parallel first!!

8. The rest of these slides are on a worksheet

9. practice Investigate the nature of the solutions (roots) for the following simultaneous equations. • 5x – 3y – 4z = 6 7.5x – 4.5y – 6z = 9 2.5x – 1.5y -2z = 3 • 7x – 6y + z = 9 6x – 7y + z = 8 5x – 8y + z = 7 This is an example of a system made up of 3 planes which are exactly the same. The system is dependent and has many solutions (any point that works in one system will work in the others). This is an example of a system of non-parallel planes that intersect at a line (a book). The system is dependent and has many solutions (any point on the line of intersection).

10. One more for practice (slightly harder…) Investigate the nature of the solutions (roots) for the following simultaneous equations. 6x – 2y – 5z = 7 9x – 3y – 7.5z = 10 72x – 24y – 60z = 14 This is an example of a system made up of 3 parallel planes (magic carpet). The system is inconsistent and has no solutions.

11. Eliminate one of these variables to decide whether you have a tent or a book Hint: You will have to make sure one of the variables has the same coefficient for each equation

12. Give a geometrical description of how the planes represented by these three equations relate toeach other. Give mathematical reasons to justify your answer.

13. Answer! You get a book! (Dependent, many solutions) You can see that they are not parallel at all (so not magic carpets or ninja style) Eliminate one of the variables. The equations you get are exactly the same, so they are along the same line and you have a book 