Question

Explain how to solve the following system of equations. What is the solution to the system?
2x + 2y + z = -5
3x + 4y + 2z = 0
x + 3y + 2z = 1

Answer 1

What methods of solving systems of equations in 3 variables have you already learned? elimination? matrices? which approach would you prefer to use here?

Answer 2

I've learned both elimination and matrices. I'd prefer to solve it using elimination though. I'm more comfortable using that to solve problems then matrices.

Answer 3

Take a look at the 3 equations. Which of the variables, x, y or z, would be easiest to eliminate by addition / subtraction?

Answer 4

y?

Answer 5

no wait z!

Answer 6

Yes, z is a much, much better choice.

By what would you have to multiply the first equation so that the coeff. of z is -2?

Answer 7

-1 right?

Answer 8

We';ve agreed that we want to eliminate variable z. Look again at the first equation and then decide whether multiplying this equation by -1 or by -2 would be better.

Answer 9

2x + 2y + z = -5
3x + 4y + 2z = 0
x + 3y + 2z = 1

Answer 10

oh since the first equation is 1 it would be -2 my b. I was looking at the other two.

Answer 11

Right. Please multiply the first equation (only) by -2.

Answer 12

(2x + 2y + z = -5)(-2)
3x + 4y + 2z = 0
x + 3y + 2z = 1

Answer 13

-4x -4y -2z = 10

Answer 14

-4x- 4y -2z = 10 Good
3x + 4y + 2z = 0
x + 3y + 2z = 1

Now we're ready to elim. z. First, add the 1st 2 equations together. Cancel the z terms.

Next, add the 1st and 3rd eq'ns together. cancel the z terms.

You get which 2 equations?

Answer 15

-1x = 10
-3x -1y = 11
right?

Answer 16

I haven't actually done the problem.
I rather expected you'd get ax + by = c and dx + ey = f

But that doesn't mean you're wrong.

want to check those calculations once more?

combine eq'ns 1 and 2.
Next, combine eq'ns 1 and 3

I will do this myself as you do it .

Answer 17

Mine are the same as yours, except in your very last equation I obtained -3x + y =11. You got -3x - y = 11. Which do you believe is correct?

Answer 18

I think mine is because you're adding a positive 3y to a negative 4y so there's going to be a -1y left over...right?

Answer 19

Let's go with yours.

-x=10
-3x - y = 11

Are we in agreement?

Answer 20

Yep

Answer 21

Pls solve the first equation for x. x=?
Next, substitute this numerical value for x in the 2nd equation. What do y ou get?

Answer 22

Apologies for the long delay. OpenStudy went haywire for 4-5 minutes.

Answer 23

It's fine.
I got x = -10
then I plugged it into the second equation and got y = 19

Answer 24

Great. Now, how would you find the value of z that satisfies this system of equations?

Answer 25

plug x and y into the first equation right?

Answer 26

Any of the original 3 equations would do the trick. You know x and y values, so only z would be unknown. Please do choose the 1st of the original 3 eq'ns. Subst. the known values of x and y and find z.

Answer 27

ok so x = -10
y = 19 and I got
z= -23

Answer 28

Those are my solutions?

Answer 29

Substitute your x=-10, y =19 and z = -23 into either the 2nd or the 3rd equation.
If the resulting equation is true, your solution set is correct.

Answer 30

Alright, thank you soo much!

Answer 31

Thanks for your persistence, Emma. It was a joy to work with you. Sorry for the long delays you experienced.

Answer 32

Don't worry about it! It was worth it haha