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Use Gaussian Elimination on the following system of equations to solve for z in terms of y. (Do this by first eliminating x)

x+2y-z = 2

-x+3y+2z = 13

2x-y+3z = 19


(A) z = 3+y

(B) z = -2y

(C) z = y-3

(D) z = y+2

User Carolanne
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Answer:

To solve the system of equations using Gaussian elimination, we'll perform row operations to eliminate the variable x. The given system of equations is:

x + 2y - z = 2

-x + 3y + 2z = 13

2x - y + 3z = 19

First, let's use equations 1 and 2 to eliminate x. Add equation 1 to equation 2:

(1) + (2): (x + 2y - z) + (-x + 3y + 2z) = 2 + 13

Simplifying, we get:

5y + z = 15 ...........(4)

Now, we have two equations with variables y and z:

5y + z = 15

2x - y + 3z = 19

Next, we'll use equations 3 and 4 to eliminate x. Multiply equation 3 by 2 and equation 4 by 5:

2 * (3): 4x - 2y + 6z = 38

5 * (4): 25y + 5z = 75

Now, let's add the two equations:

(2 * (3)) + (5 * (4)): (4x - 2y + 6z) + (25y + 5z) = 38 + 75

Simplifying, we get:

4x + 23y + 11z = 113 ...........(5)

Now we have three equations with variables y and z:

5y + z = 15

4x + 23y + 11z = 113

To eliminate x, we'll multiply equation 4 by 4:

4 * (4): 20y + 4z = 60

Now, let's subtract equation 5 from this new equation:

(4 * (4)) - (5): (20y + 4z) - (4x + 23y + 11z) = 60 - 113

Simplifying, we get:

-4x - 3y - 7z = -53 ...........(6)

Now we have two equations with variables y and z:

-4x - 3y - 7z = -53

5y + z = 15

We can solve this system of equations using Gaussian elimination. Let's eliminate y by multiplying equation 6 by 5 and equation 4 by 3:

5 * (6): -20x - 15y - 35z = -265

3 * (4): 15y + 3z = 45

Now, add the two equations:

(5 * (6)) + (3 * (4)): (-20x - 15y - 35z) + (15y + 3z) = -265 + 45

Simplifying, we get:

-20x - 32z = -220 ...........(7)

Now we have two equations with variables x and z:

-20x - 32z = -220

5y + z = 15

To eliminate z, we'll multiply equation 7 by 5 and equation 4 by 32:

5 * (7): -100x - 160z = -1100

32 * (4): 160y + 32z = 480

Now, add the two equations:

(5 * (7)) + (32 * (4)): (-100x - 160z

User Ben Everard
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