Question

Solve the following system of equations.

2x – y + z = –3
2x + 2y + 3z = 2
3x – 3y – z = –4

Answer 1

@whpalmer4 can you help me please? :)

Answer 2

do you know matrices?

Answer 3

no i dont...sorry

Answer 4

solve by cancellation.......

Answer 5

yeah, matrices is probably the straightforward way to do this, but I never do them so we'll have to do elimination.

Answer 6

\[2x-y+z=-3\]\[2x+2y+3z=2\]\[3x-3y-z=-4\]

I would first subtract the first equation from the second equation to get one in just y and z. Then I would multiply the second equation by 3 and the second equation by 2, subtract and get another equation in just y and z. Now do elimination on those two and you get a solution for y and z, and you back substitute to find x.

Answer 7

why don't you try that and see how far you get? I'll help you through it if you get stuck and show me your work...

Answer 8

@juliaann how are we doing?

Answer 9

I'm lost...my work isnt even worth posting....can you help?

Answer 10

looking at the equations again, I'm going to suggest a different strategy. First we add the 1st and 3rd equations:

\[2x-y+z=-3\]\[3x-3y-z=-4\]-------------------\[(2+3)x-(1+3)y+(1-1)z = -7\]or\[5x-4y=-7\]

Agreed?

Answer 11

yes im following!

Answer 12

Okay, now we need to find another combination that will eliminate z so we have two equations in just x and y. Any suggestions?

Answer 13

I of course did the easiest one :-)

Answer 14

lol I'm understanding but im just not sure what so suggest

Answer 15

well, how about we combine the 2nd and 3rd equations to eliminate z. what do we have to do to make them eliminate z?

Answer 16

what are the coefficients of z in the 2nd and 3rd equations?

Answer 17

2x and 3x?

Answer 18

Nah...

you know what coefficients are, right? just the numbers (and sign) in front of the variable.

second equation is
\[2x+2y+3z=2\]
coefficient of z =

third equation is
\[3x−3y−z=−4\]
coefficient of z =

Answer 19

2 and 3?

Answer 20

"3z"
coefficient of z =

Answer 21

3, right?

Answer 22

"-z"
coefficient of z =

Answer 23

yess

Answer 24

oh, dear, openstudy is showing my posts in the wrong order! :-(

Answer 25

coefficient of z in "3z" = 3
coefficient of z in "-z" = -1

does that make sense?

Answer 26

yes that makes sense

Answer 27

(1, 3, 2)

(–1, 3, 2)

(1, –3, 2)

(1, 3, –2)

Answer 28

these are the answer choices A, B, C, D

Answer 29

okay. we need to do something to the 2nd and 3rd equations so that the coefficients will be 3 and -3 (or -3 and 3) so that when we add the equations together, the z term disappears leaving us with another equation in just x and y

Answer 30

as one of our coefficients is 3 and the other is -1, I suggest we just multiply the equation with the -1 by 3!

\[3*3x−3*3y−3*1z=−4*3\]
that gives us:
\[9x - 9y -3z = -12\]\[2x+2y+3z=2\]--------------------
\[(9+2)x +(-9+2)y + (-3+3)z = -12+2\]\[11x -7y = -10\]

So now we have two equations in just x and y:

\[5x-4y=-7\]\[11x-7y=10\]

Answer 31

are you still here? @whpalmer4

Answer 32

Yeah, I'm trying to find a mistake I made somewhere :-(

Answer 33

pipe up if you see it :-)

Answer 34

okay ill take a look!

Answer 35

Ah, I just didn't type a minus sign when copying!

So we have two equations in just x and y:
\[5x-4y = -7\]\[11x-7y=-10\]

We can combine them to eliminate a variable. I'm going to eliminate y because I can multiply by smaller numbers than if I try to eliminate x. I'll multiply the first equation by 7 and the second by -4:

\[7*5x - 7*4y = -7*7\rightarrow 35x-28y = -49\]\[-4*11x-7(-4)y = (-4)(-10)\rightarrow -44x+28y = 40\]
We add our two new equations
\[35x-28y = -49\]\[-44x+28y=40\]-----------------\[-9x=-9\]\[x=1\]hooray!

Answer 36

now we work backwards to find y and z.

\[5x-4y = -7\]\[5(1)-4y= -7\]\[5+7 = 4y\]\[12=4\]\[y=3\]

another cheer goes up from the crowd :-)

now we know x and y, let's find z

\[2x−y+z=−3\]\[2(1) -(3) + z = -3\]\[2-3+z=-3\]\[z = -3+1\]\[z = -2\]
And the crowd goes wild :-)

Answer 37

youre insanely good at math!! thankyou so much for your time :)

Answer 38

so now we really should check our answers in all 3 equations... if we have multiple equations, we have to try them all to be sure the answer is correct, as it is possible to come up with answers that work for some but not all of the equations.

Answer 39

our answers were (1,3,-2)

2(1)-3+(-2) = -3
2-3-2 = -3
-3=-3
1st eq OK

2(1)+2(3)+3(-2) = 2
2+6-6 = 2
2=2
2nd eq OK

3x-3y-z=-4
3(1)-3(3)-(-2)=-4
3-9+2=-4
-6+2=-4
-4=-4
3rd eq OK