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Find the ordered triple (p,q,r) that satisfies the following system:

p - 2q = 3

q - 2r = -2 + q

p + r = 9 + p

 May 31, 2023
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To solve this system, we can use algebraic manipulation to eliminate variables and solve for one variable in terms of the others.

First, let's simplify the second equation by moving the q term to the left-hand side:

q - 2r - q = -2

Simplifying, we get:

-q - 2r = -2

Now we can eliminate q by adding the first equation to this simplified second equation:

(p - 2q) + (-q - 2r) = 3 - 2

Simplifying, we get:

p - 3q - 2r = 1

Next, we can use the third equation to eliminate r by solving for p in terms of r and substituting into the third equation:

p + r = 9 + p

Subtracting p from both sides, we get:

r = 9

Now we can substitute this value of r into the equation we just derived:

p - 3q - 2r = 1

p - 3q - 2(9) = 1

Simplifying, we get:

p - 3q = 19

Finally, we can use the first equation to solve for p in terms of q:

p - 2q = 3

p = 2q + 3

Substituting this expression for p into the equation we just derived, we get:

(2q + 3) - 3q = 19

Simplifying, we get:

-q = 16

Dividing both sides by -1, we get:

q = -16

Now we can use the expression we derived earlier to solve for p:

p = 2q + 3 = 2(-16) + 3 = -29

Therefore, the ordered triple that satisfies the system is (-29, -16, 9).

 May 31, 2023

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