Question

Solve the system of equations.

3x - 4y + z = 21
-2x - 3y + z = -1
4x = 16
(x = 4)

Answer 1

@ybarrap - please help me.

Answer 2

Ok. Let's start. Do you see any equation that has only ONE variable?

Answer 3

@ybarrap - Yes. 4x = 16; x = 4.

Answer 4

Great! Now we 1/3 done.

Next we plug this value into the 1st equation. What do you get? You should have one equation with 2 variables left, because the \(x\) variable goes away.

Just work with the 1st equation for now.

Answer 5

3x - 4y + z = 21
3(4) - 4 y + z = 21
12 - 4y + z = 21

Answer 6

Nice. Now do exactly the same thing with the second equation. Plug in x=4 and you should again have an equation with two variable left.

Just work with the second equation now.

Answer 7

-2x - 3y + z = -1
-2(4) - 3y + z = -1
-8 - 3y + z = -1

Answer 8

Looking good so far. This is what we have at this point:
$$
12 - 4y + z = 21\\
-8 - 3y + z = -1
$$
We want it so that one variable and ONLY one is to the left of the equal sign. We can choose any of these two equations. Let's choose the second.

We want to eliminate that -8 and -3y from the left side. This will leave the variable \(z\) alone on the left side of the equation.

How can I eliminate the -8? One way is to add +8 to the left side. Right?

But if we do this to one side of the equation, we also need to to this to the right side.

So go ahead and add +8 to the left side and the right side of the second equation that I just mentioned.

Answer 9

-8 - 3y + z = -1
-3y + z = 7

Answer 10

Next we use the same technique to eliminate the -3y. So, we add +3y to the left and of course to the right hand side.

What do you get?

Answer 11

-3y + z = 7
z = 3y + 7

Answer 12

Great! We're almost done.

We now have an equation for \(z\).

Let's look back at the two equations we had with only \(y\) and \(z\):
$$
12 - 4y + z = 21\\
-8 - 3y + z = -1
$$
We've been working with the second equation. Now we attack the 1st equation.

You used the second equation to come up with
$$
z = 3y + 7
$$

Now plug this into the 1st equation here.

What do you get?

Answer 13

12 - 4y + z = 21
12 - 4y + (3y + 7) = 21
-y + 19 = 21

-y + 19 = 21
- y = 2
y = -2

Answer 14

Very awesome!

Now we have x=4, y=-2 and now just need z.

You can use the the value for \(y\) and the equation you had for \(z\) above to complete this problem:
$$
z = 3y + 7
$$
When you are done, plug these values into your original equations and make sure that they work:
$$
3x - 4y + z = 21\\
-2x - 3y + z = -1\\
4x = 16
$$
That is, when plugging in x,y and z in the 1st equation, you should get 21. Then -1 for the second and 16 for the last.

That's it!

Answer 15

do you guys need help

Answer 16

z = 3y + 7
z = 3(-2) + 7
z = (-6) + 7
z = 1

Check:
3x - 4y + z = 21
3(4) - 4(-2) + 1 = 21
12 -(-8) + 1 = 21
12 + 8 + 1 = 21
21 = 21

-2x - 3y + z = -1
-2(4) - 3(-2) + 1 = -1
-8 + 6 + 1 = -1
-1 = -1

4x = 16
4(4) = 16
16 = 16

Answer 17

Perfect!!

Answer 18

@ybarrap - Thank you so much!

Answer 19

You are very welcome. Keep it up!