Intermediate algebra - "Solve each systems"

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January 23rd 2010, 09:26 PM
Justiceapple

Intermediate algebra - "Solve each systems"

Hi, I've been trying to solve this for over an hour now by extending the method of elimination like the textbook says. Very frustrated. (Headbang)
Please help. (Crying)

Solve each systems. Express the solution in the form (x,y,z,w).

x+y+z-w=5
2x+y-z+w=3
x-2y+3z+w=18
-x-y+z+2w=8

The answer is (2,1,5,3)
I don't know how to get there.
January 23rd 2010, 10:42 PM
Prove It

Quote:

Originally Posted by Justiceapple

Hi, I've been trying to solve this for over an hour now by extending the method of elimination like the textbook says. Very frustrated. (Headbang)
Please help. (Crying)

Solve each systems. Express the solution in the form (x,y,z,w).

x+y+z-w=5
2x+y-z+w=3
x-2y+3z+w=18
-x-y+z+2w=8

The answer is (2,1,5,3)
I don't know how to get there.

$x + y + z - w = 5$
$2x + y - z + w = 3$
$x - 2y + 3z + w = 18$
$-x - y + z - 2w = 8$ .

Apply $R_2 - 2R_1 \to R_2, R_3 - R_1 \to R_3, R_4 + R_1 \to R_1$ .

$x + y + z - w = 5$
$-y - 3z + 3w = -7$
$-3y + 2z + 2w = 13$
$2z - 3w = 13$

Apply $-R_2 \to R_2$

$x + y + z - w = 5$
$y + 3z - 3w = 7$
$-3y + 2z + 2w = 13$
$2z - 3w = 13$

Apply $R_3 + 3R_2 \to R_3$

$x + y + z - w = 5$
$y + 3z - 3w = 7$
$11z - 7w = 34$
$2z - 3w = 13$

Interchange $R_3$ and $R_4$ .

$x + y + z - w = 5$
$y + 3z - 3w = 7$
$2z - 3w = 13$
$11z - 3w = 13$

Apply $\frac{1}{2}R_3$

$x + y + z - w = 5$
$y + 3z - 3w = 7$
$z - \frac{3}{2}w = \frac{13}{2}$
$11z - 3w = 13$

Apply $R_4 - 11R_3$

$x + y + z - w = 5$
$y + 3z - 3w = 7$
$z - \frac{3}{2}w = \frac{13}{2}$
$\frac{27}{2}w = -\frac{117}{2}$

So $w = \frac{2}{27}\left(-\frac{117}{2}\right)$

$w = -\frac{13}{3}$ .

$z - \frac{3}{2}w = \frac{13}{2}$

$z - \frac{3}{2}\left(-\frac{13}{3}\right) = \frac{13}{2}$

$z + \frac{13}{2} = \frac{13}{2}$

$z = 0$ .

$y + 3z - 3w = 7$

$y + 3(0) - 3\left(-\frac{13}{3}\right) = 7$

$y + 13 = 7$

$y = -6$ .

$x + y + z - w = 5$

$x - 6 + 0 + \frac{13}{3} = 5$

$x - \frac{5}{3} = 5$

$x = \frac{20}{3}$ .

To check, sub into original Eq 2:

$2x+y-z+w = 2\left(\frac{20}{3}\right) - 6 - 0 - \frac{13}{3}$

$= \frac{40}{3} - \frac{18}{3} - \frac{13}{3}$

$= \frac{9}{3}$

$= 3$ as required.

So $(x, y, z, w) = \left(\frac{20}{3}, 6, 0, -\frac{13}{3}\right)$ .

The solution your text book gives is incorrect.
January 23rd 2010, 11:08 PM
Justiceapple

Oh no, you wrote down the problem wrong :(

The R4 is $-x-y+z+2w=8$ . It's +2w.

But I think I kind of get what to do now. Let me work on this problem first thing in the morning tomorrow. My head hurts from studying Logic for another class right now....

Thank you so much. When I saw that you replied, I screamed like I won a lottery or something. I appreciate your kindness very much.

P.S. I studied Math for 6 hours today ...
January 23rd 2010, 11:11 PM
Prove It

Oh hahahaha, yes I copied it down wrong.

But you see the basic structure now.

Try to solve yours the same way.
January 23rd 2010, 11:14 PM
VonNemo19

Quote:

Originally Posted by Justiceapple

Oh no, you wrote down the problem wrong :(

(Rofl) That's pretty funny.
January 23rd 2010, 11:36 PM
Prove It

Quote:

Originally Posted by VonNemo19

(Rofl) That's pretty funny.

So I think the score is Prove It -1, Von Nemo -2 (Rock)