\(Find the ordered triple $(p,q,r)$ that satisfies the following system: \begin{align*} p - 2q &= 3, \\ q - 2r &= -2, \\ p + r &= 9. \end{align*}\)
We can solve this system of equations using the substitution method.
Equation 1: p - 2q = 3
Equation 2: q - 2r = -2
Equation 3: p + r = 9
First, we can solve Equation 1 for p:
p - 2q = 3 p = 3 + 2q
Now, we can substitute this expression for p in Equation 3:
(3 + 2q) + r = 9 2q + r = 6
Next, we can solve Equation 2 for q:
q - 2r = -2 q = 2r - 2
Now, we can substitute this expression for q in Equation 2:
(2r - 2) - 2r = 6 -4r = 8 r = -2
Now that we know the value of r, we can substitute it into Equation 2 to find q:
q - 2(-2) = -2 q = -2
Finally, we can substitute the values of q and r into Equation 1 to find p:
p - 2(-2) = 3 p = 3
Therefore, the ordered triple that satisfies the system of equations is (3, -2, -2).
To check our answer, we can substitute these values into each equation and see that they all hold true.
p - 2q = 3 3 - 2(-2) = 3 3 = 3 q - 2r = -2 -2 - 2(-2) = -2 -2 = -2 p + r = 9 3 + (-2) = 9 9 = 9
As you can see, the ordered triple (3, -2, -2) satisfies all three equations. Therefore, it is the solution to the system.
p - 2q = 3,
q - 2r = -2,
p + r = 9, solve for p, q, r
Use substitutions to get:
p==7, q==2, r==2
The system of equations is:
p - 2q = 3 q - 2r = -2 p + r = 9
We can solve this system using Gaussian Elimination.
First, we can add the first and second equations to get:
p - 2q + q - 2r = 1
This simplifies to:
-q - 2r = 1
Now, we can subtract the third equation from this equation to get:
-q - 2r - (p + r) = 1 - 9
This simplifies to:
-2q = -8
Therefore, q = 4.
Now, we can substitute this value into the second equation to get:
4 - 2r = -2
This simplifies to:
2r = 6
Therefore, r = 3.
Finally, we can substitute these values into the first equation to get:
p - 2(4) = 3
This simplifies to:
p = 5
Therefore, the ordered triple that satisfies the system of equations is (5, 4, 3).
