system of equations (thank you in advance!!!!)

Question

show your work for which of the following is a solution to the system of equations

3x − 3y + 6z = 6

x + 2y − z = 5

5x − 8y + 13z = 7

a. (5, 3, 0)

b. (−2, 6, 5)

c. (−z + 3, z + 1, z)

Answer 1

First, to aid in communication, I'll label each equation:

Eq 1: 3x − 3y + 6z = 6

Eq 2: x + 2y − z = 5

Eq 3: 5x − 8y + 13z = 7

Step 1: I'm going to simplify Eq 1 by dividing both sides by 3 which gives us:

Eq 1A: x − y + 2z = 2

Step 2.: I'm going to eliminate the x variable by multiplying Eq 1A by -5 (resulting in Eq 1B) and adding it to Eq 3 and then secondly multiplying Eq 2 by -5 (resulting in Eq 2A) and adding it to Eq 3:

Eq 1B: -5x + 5y - 10z = -10

Eq 3: 5x − 8y + 13z = 7

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Eq 4: -3y + 3z = -3

Eq 2A: -5x - 10y + 5z = -25

Eq 3: 5x − 8y + 13z = 7

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Eq 5: -18y + 18z = -18

Now, notice that Eq 4 and Eq 5 are really the same equations. That means we will not get a numeric answer to the problem. We have a situation where the 3 planes (each equation is a plane in 3-D space) intersect at a line instead of at a point.

So, the best we can do is solve the equations in terms of the others.

Taking Eq 4 and dividing both sides by 3 we get:

Eq 4A: -y + z = -1

Solving for y we get: y = z + 1

Taking Eq 2 and plugging "z + 1" in for "y", we get:

Eq 2B: x + 2(z + 1) − z = 5

solving for x we get:

x + 2z + 2 − z = 5

x + z + 2 = 5

x + z = 3

x = 3 - z

So, the (x, y, z) solution is (3 - z, z + 1, z) (answer c)

BUT, answer b) also works because it satisfies the criteria I specified above (letting z = 5, y = 6 and x = -2

Answer 2

Option C is the solution

Option A does not work.

First step, do -3* eq 2 + eq 1 and -5 * eq2 + eq3

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-9y + 9z = -9

x + 2y - z=5

-18y +18z = -18

equations 1 and 3 are the same.

The system becomes:

y - z = 1 <-- divides equation #1 by -9

x + 2y - z = 5

Per option C, variable Z is the free variable

y = z+1

substitutes into the original second equation:

5 = x + 2y - z = x + 2(z+1) - z = x + 2z + 2 - z = x + 2 + z

Therefore, 3 = x + z, so x = 3-z = -z + 3

The general solution is the ordered triplet ( x = -z +3, y = z+1, z)

Notice that for z=5 gives the particular solution of option B