\(Find the ordered triple $(p,q,r)$ that satisfies the following system: \begin{align*} p - 2q &= 3, \\ q - 2r &= -2, \\ p + r &= 9. \end{align*}\)

Guest Jun 26, 2023

#1**0 **

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.

Guest Jun 26, 2023

#3**0 **

p - 2q = 3,

q - 2r = -2,

p + r = 9, solve for p, q, r

Use substitutions to get:

**p==7, q==2, r==2**

Guest Jun 26, 2023

#4**0 **

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).**

Guest Jun 26, 2023