Find the ordered triple (p,q,r) that satisfies the following system:
p - 2q = 3
q - 2r = -2 + q
p + r = 9 + p
We can start by simplifying the second equation:
q - 2r = -2 + q
Subtracting q from both sides, we get:
-2r = -2
Dividing by -2, we get:
r = 1
Now we can substitute r = 1 into the third equation:
p + r = 9 + p
p + 1 = 9 + p
Subtracting p from both sides, we get:
1 = 9
This is a contradiction, so there are no solutions to the system. Therefore, there is no ordered triple that satisfies the given equations.
To find the ordered triple (p,q,r) that satisfies the given system of equations, let's solve the system step-by-step.
Step 1: Solve the first equation for p
p - 2q = 3
Add 2q to both sides:
p = 3 + 2q
Step 2: Substitute the value of p from Step 1 in the third equation
p + r = 9 + p
(3 + 2q) + r = 9 + (3 + 2q)
3 + 2q + r = 12 + 2q
Subtract 2q from both sides:
3 + r = 12
Step 3: Solve the equation from Step 2 for r
3 + r = 12
Subtract 3 from both sides:
r = 12 - 3
r = 9
Step 4: Substitute the value of r from Step 3 into the second equation
q - 2r = -2 + q
q - 2(9) = -2 + q
q - 18 = -2 + q
Add 2 - q to both sides:
q - q = -2 + 18
0 = 16
Since 0 does not equal 16, this implies that the system of equations has no solution.
Therefore, there is no ordered triple (p,q,r) that satisfies the given system of equations.