https://file.aiquickdraw.com/mathbot/file/1728662158123f3teetms.jpg

The system of equations shown in the image is:

\[
\begin{aligned}
1. \quad 2p - q + 4r &= 11 \\
2. \quad p + 2q - 6r &= -11 \\
3. \quad 3p - 2q - 10r &= 11
\end{aligned}
\]

### Step 1: Solve one equation for a variable
Let's solve the second equation for \( p \):

\[
p = -2q + 6r - 11
\]

### Step 2: Substitute \( p \) in the other equations
Now, substitute this expression for \( p \) into the first and third equations:

#### Substituting into the first equation:

\[
2(-2q + 6r - 11) - q + 4r = 11
\]

Simplify:

\[
-4q + 12r - 22 - q + 4r = 11
\]

\[
-5q + 16r = 33
\quad \text{(Equation 4)}
\]

#### Substituting into the third equation:

\[
3(-2q + 6r - 11) - 2q - 10r = 11
\]

Simplify:

\[
-6q + 18r - 33 - 2q - 10r = 11
\]

\[
-8q + 8r = 44
\quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations
We now have the system:

\[
\begin{aligned}
-5q + 16r &= 33 \\
-8q + 8r &= 44
\end{aligned}
\]

Multiply the second equation by 2 to eliminate \( r \):

\[
-16q + 16r = 88
\]

Subtract the first equation from this:

\[
(-16q + 16r) - (-5q + 16r) = 88 - 33
\]

Simplify:

\[
-11q = 55
\]

\[
q = -5
\]

### Step 4: Solve for \( r \)
Substitute \( q = -5 \) into the first reduced equation:

\[
-5(-5) + 16r = 33
\]

\[
25 + 16r = 33
\]

\[
16r = 8 \quad \Rightarrow \quad r = \frac{1}{2}
\]

### Step 5: Solve for \( p \)
Substitute \( q = -5 \) and \( r = \frac{1}{2} \) into the equation for \( p \):

\[
p = -2(-5) + 6\left(\frac{1}{2}\right) - 11
\]

\[
p = 10 + 3 - 11 = 2
\]

### Final Solution:
The solution to the system is:

\[
p = 2, \quad q = -5, \quad r = \frac{1}{2}
\]

Would you like more details or have any questions about this process? Here are some related questions to explore:

1. How do you solve systems of equations using substitution?
2. What other methods can be used to solve a system of equations (e.g., elimination)?
3. How does solving a system with three variables differ from two-variable systems?
4. Why is it important to check solutions in all original equations?
5. Can this system be solved using matrices or determinants?

**Tip**: Always substitute your solution back into the original system of equations to verify that it satisfies all the equations.

Explore how to solve a system of three linear equations using the substitution method. The problem involves algebraic concepts such as solving for variables, simplifying expressions, and applying the elimination method to find the solution for p, q, and r. Suitable for students in Grades 9-12.

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