Starter mix and match the following Inconsistent Exactly

Starter – mix and match the following Inconsistent Exactly the same planes Dependent Like flying carpets Parallel planes Many solutions No solutions

Mix and match answers Dependent Many solutions Exactly the same planes Inconsistent No solutions Parallel planes Like flying carpets

Recap from yesterday 1) Unique – one solution 2) Same planes (dependent – many solutions) 3) All 3 parallel – magic carpets (inconsistent – no solutions) 4) 2 planes parallel and one cuts through them – ninja styles (inconsistent – no solutions)

Order to do things: • Always check for parallel planes first • If all 3 are parallel, you have magic carpets. • If only 2 are parallel, you have ninja styles • But, you might get solutions that are neither of these…

5) Planes intersect along a common line – The book • Many solutions • All three planes intersect along a common line. • We can say the equations are dependent. The answers you will get for x, y and z can change, they will depend on what one of the variables is.

5) The book Dependent 6 x + 9 y - 27 z = 48 6 x + 2 y - 2 z = 24 6 x - 12 y + 48 z = -24 Three planes intersect along a common line, so you get many solutions

5) The book Dependent 6 x + 9 y - 27 z = 48 6 x + 2 y - 2 z = 24 6 x - 12 y + 48 z = -24 Three planes intersect along a common line. To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. In this case, the x variable is the easiest to eliminate as they have the same coefficient

5) The book Dependent To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. 6 x + 9 y - 27 z = 48 6 x + 2 y - 2 z = 24 6 x - 12 y + 48 z = -24 7 y - 25 z = 24 To eliminate the 6 x, we want to subtract one equation from the other

5) The book Dependent To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. 6 x + 9 y - 27 z = 48 6 x + 2 y - 2 z = 24 6 x - 12 y + 48 z = -24 7 y - 25 z = 24 14 y - 50 z = 48 To eliminate the 6 x, we want to subtract one equation from the other Then take the other 2 equations and eliminate the 6 x by subtracting one equation from the other Remember, these are the lines that the pairs of equations intersect along. If these lines are the same, then you have a book

5) The book Dependent To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. 6 x + 9 y - 27 z = 48 6 x + 2 y - 2 z = 24 Remember, these are the lines that the pairs of equations intersect along. If these lines are the same, then you have a book 6 x - 12 y + 48 z = -24 So are they the same line? 7 y - 25 z = 24 14 y - 50 z = 48 The second equation can be divided through by 2 to give the first equation. So we have a book.

6) The lines where the planes intersect are parallel to each other – The tent • No solutions • These planes never intersect • We can say the equations are inconsistent.

6) The tent 15 x + 5 y + 20 z = 500 5 x + 5 y + 5 z = 375 -5 x + 5 y + -10 z = 0 Starts out like a book, but with the bottom plane moved up. The 3 planes no longer intersect along the same line The intersections of 2 planes at a time are all parallel (Tent)

6) The tent To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. In this case, the y variable is the easiest to eliminate as they have the same coefficient 15 x + 5 y + 20 z = 500 5 x + 5 y + 5 z = 375 -5 x + 5 y + -10 z = 0 10 x + 15 z = 125 To eliminate the 5 y, we want to subtract one equation from the other

6) The tent To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. In this case, the y variable is the easiest to eliminate as they have the same coefficient 15 x + 5 y + 20 z = 500 5 x + 5 y + 5 z = 375 -5 x + 5 y + -10 z = 0 10 x + 15 z = 125 + 15 z = 375 To eliminate the 5 y, we want to subtract one equation from the other Then take the other 2 equations and eliminate the 6 x by subtracting one equation from the other

6) The tent To find where pairs of planes intersect, we take 2 planes at a time and try to eliminate one of the variables. 15 x + 5 y + 20 z = 500 5 x + 5 y + 5 z = 375 So this is a tent! Because the equations are the same except for the constant -5 x + 5 y + -10 z = 0 10 x + 15 z = 125 + 15 z = 375 Remember, these are the lines that the pairs of equations intersect along. If these lines are the same, then you have a book If they are parallel to each other, you have a tent.

Recap from today: If the lines are not parallel: • Eliminate one of the variables. • This gives you the equation of where pairs of your planes intersect. - If these lines are the same, you have a book (many solutions and dependent) - If these lines are parallel, you have a tent (no solutions and inconsistent)

Practice 1 • Eliminate one variable from this set of equations and determine whether you have a book or a tent

Practice answer • That was a book (the lines where the pairs of planes intersect are the same, so we have many solutions and they are dependent) It isn’t obvious whether these are the same equation or parallel, so we can put them in the form y = mx + c

Practice 2 • Eliminate one variable from this set of equations and determine whether you have a book or a tent

Answers to practice 2 • Inconsistent (no solutions)

Recap the steps: You have 3 equations, what type of solution are they? If it is not unique, complete the following: 1) 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) 2) If two of them are parallel, but the third isn’t, then you have ninja styles (inconsistent and no solutions) 3) 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.

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)

Quick recap