Question

x + 2y - z = -1
x - 4y - z = -4
2x - 2y + 3z = 15
solve using matrices

Answer 1

@mathmath333 i think you helped yesterday but my computer went down. if you could help again thatd be great :)

Answer 2

can you write down the matrix ? i.e. order the variables alphabetical (easier to remember what is what if you do that, then write the coefficients in matrix form

what do you get ?

Answer 3

1 2 -1 x -1
1 -4 -1 X y = -4
2 -2 3 z 15

Answer 4

ok, but I would leave out the multiply sign (it looks like an "x".

One way to solve is find the inverse of the matrix (tedious for a 3 x 3)
another is to use "row reduction" (not too bad)
What techniques have you been studying ?

Answer 5

matrices but i dont understand it

Answer 6

yes, but there are a few different ways. *any* idea how they want you to solve this ?
If not, we can use row reduction.

Answer 7

no specific way

Answer 8

ok, then let's use row reduction.

to start, can you answer this. IF you had the matrix
1 2 -1 x 9
-1 1 -4 y = 8
-0 0 1 z 2

do you know what z is ?

Answer 9

no i dont

Answer 10

to answer, change the last row "back" into the original equation
(remember the numbers are the coefficients for x y and z )

Answer 11

how would you change it back?

Answer 12

how did you write the matrix? you "ignored" the variable names, and just wrote the numbers
example, the first equation
x + 2y - z = -1

became the first row: 1 2 -1 -1 (I left out the unknown)

Answer 13

to go back, put in the x, y and z in the last row of my made-up matrix:
0 0 1 2

Answer 14

x + y + z = 2 ?

Answer 15

that would be if had the row 1 1 1 2
but you have 0 0 1 2

Answer 16

0x + 0y + 1z = 2

Answer 17

yes, and we can simplify that. zero times anything is zero.

Answer 18

ok

Answer 19

in other words, you have no x's and no y's

Answer 20

and get
z= 2

Answer 21

oh ok. so how would i do that with the equations up top

Answer 22

that is the idea with "reduction". It's a way to change the matrix so we have a bunch of zeros except in one place. (Like the last row of my example)

Answer 23

oh. i get your example but not mine

Answer 24

yes, but now we can tackle your problem
1 2 -1 -1
1 -4 -1 -4
2 -2 3 15

the idea is we want to change the "leading number" (the first number) in each row (except the first row) into zero.

to do this we multiply the top row by a number so that its first number (the 1)
when added to the second row, gives us zero.

so off to the side, we say: the second row has a 1 that we want to change to zero. We do this by adding -1. I need to multiply the top row by -1 to change the first number into -1.

Answer 25

in other words, we multiply the top row by -1 (that means multiply each number in the top row by -1. we get
-1 * (1 2 -1 -1) --> -1 -2 +1 + 1

Answer 26

now add -1 -2 +1 + 1 to the second row (number by number)
can you write down what you get ?

Answer 27

0 -2 -1 -3

Answer 28

Here is the original matrix
1 2 -1 -1
1 -4 -1 -4
2 -2 3 15

Off to the side we multiply the first row by -1 and get -1 -2 +1 +1
add that to the *second row*:

-1 -2 +1 +1
1 -4 -1 -4
--------------
<--- sum goes here

Answer 29

can you try again?

Answer 30

0 -6 0 -3

Answer 31

yes. that becomes our "new" second row.
the problem is now
1 2 -1 -1
0 -6 0 -3
2 -2 3 15

Now let's change the first 2 in the last row to a zero. what do we add to 2 to make it 0?

Answer 32

add -2

Answer 33

yes. what to we multiply the first row by so it's first number becomes -2 ?

Answer 34

1

Answer 35

1 times the first row will give you 1 2 -1 -1
you want the -2 ...rest of row...

in other words, we need to multiply the 1 by something to get -2
if it's not clear, you can say

1*x = -2
what is x ?

Answer 36

-2

Answer 37

though I should not use "x". I mean what unknown number N should we use?

Yes -2

can you write down what you get when you multiply the first row by -2?

Answer 38

the first row being 1 2 -1 ?

Answer 39

yes, but with the other -1 (on the other side of the = sign)
1 2 -1 -1

I know this is a bit mysterious...

Answer 40

-2 -4 2 2

Answer 41

yes, now add that to the third row in our matrix:
1 2 -1 -1
0 -6 0 -3
2 -2 3 15

Answer 42

0 -6 5 17

Answer 43

with row reduction we would next tackle the next column in and zero out the entries below the the second row. But in this case notice the second row already has two zeros

Answer 44

i see

Answer 45

can you write down the equation to "goes with" the second row ?

Answer 46

-6y = -3

Answer 47

can you solve for y ?

Answer 48

y=.5

Answer 49

yes.
next we tackle the bottom row which has only y and z in it.
write down the equation that you get from the third row.

Answer 50

-6y + 5z = 17

Answer 51

and we know y= 1/ 2
put in 1 /2 for y, and solve for z. can you do that ?

Answer 52

i dont understand

Answer 53

you have
-6y + 5z = 17

we already figured out y is 1/2
so replace y with 1/2
-6 * 1/2 + 5z = 17
simplify
and solve for z. do you know how ?

Answer 54

as a first step, I would multiply -6 * 0.5

Answer 55

i got 3.3

Answer 56

@phi

Answer 57

can you show your work?

Answer 58

-6 * 1/2 + 5z = 17
becomes
-3 + 5z = 17
next, add +3 to both sides.

Answer 59

1/2 + 5z = 17
add - 1/2 to both sides
5z = 17 - 1/2
5z = 16.5
divide both sides by 5 and get 3.3

Answer 60

Oh. you did 1y + 5z = 17
and replaced y with 1/2 to get 1/2 + 5z = 17
the work looks good.

but we are doing the last row, which is
0 -6 5 17
and that becomes

-6*y + 5 z = 17

with y=1/2 it becomes
-6* 1/2 + 5z = 17
now solve for z

Answer 61

can you help me im not sure how to do it

Answer 62

when you see two numbers being multiplied, it's a good idea to simplify to get one number
-6* 1/2 + 5z = 17

Answer 63

-3 + 5z = 17
5z = 17 + 3
5z = 20
z= 4

Answer 64

most excellent. we now know y = 1/2 and z = 4
now we use the top row.
care to tackle it ?

Answer 65

-6y + 5z = 17?

Answer 66

@phi

Answer 67

we already used that row
we need to solve for "x"
and we use the first row in the matrix

Answer 68

ohh so -2x + -4y + 2z = 2

Answer 69

yes, except I noticed I pasted in the wrong matrix.
It should be
1 2 -1 -1
0 -6 0 -3
0 -6 5 17

we used the second and third rows to find y and then z

Answer 70

oh ok

Answer 71

so does that change the equation

Answer 72

we have to use the correct matrix. The one I pasted with -2 -4 2 2 was just *wrong* we want to use
1 2 -1 -1
0 -6 0 -3
0 -6 5 17

Answer 73

ok

Answer 74

so we use
1 2 -1 -1

can you find "x" ?

Answer 75

x+2y -z =-1

Answer 76

yes. and now replace y with 1/2 and z with 4

Answer 77

-5.5

Answer 78

go slower. one step at a time. first, *just* replace y and z with their numbers in
x+2y -z =-1
what do you get ?

Answer 79

x + 1/2 - 4=-1

Answer 80

ok but you should not leave out the coefficients.
when you start with
x+2y -z =-1
you replace the y with 1/2 but *you keep the 2*

Answer 81

oh okay

Answer 82

in other words, the equation
x + 2y - z = -1
says x plus *TWO* y's take away z is -1
we need to multiply the y by 2

Answer 83

qnd get 4y?

Answer 84

I meant when we start with
x + 2y - z = -1
and replace y with 1/2 we write
x + 2*1/2 - z = -1
and replace z with 4:
x + 2* 1/2 - 4 = -1

ok?

Answer 85

ok

Answer 86

can you finish ?
x + 2* 1/2 - 4 = -1

Answer 87

X=2

Answer 88

yes. we have solved the problem
if you multiply the original matrix times the vector
2
0.5
4
you will get the right hand side vector
-1
-4
15

Answer 89

also, each of the 3 original equations will be true if we use the values
x=2, y=1/2 and z=4

Answer 90

If you have time, Khan goes into detail how to do this type of problem
https://www.khanacademy.org/math/precalculus/precalc-matrices/reduced_row_echelon/v/matrices-reduced-row-echelon-form-1