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Let's solve the given system of equations using the elimination method. The equations are:

1. \( p + q - 40 = 0 \)
2. \( 7p + 3q = 180 \)

### Step 1: Simplify the first equation
Rearrange the first equation to express it in standard form:
\[ p + q = 40 \]

### Step 2: Write both equations
Now, the system of equations is:
1. \( p + q = 40 \)
2. \( 7p + 3q = 180 \)

### Step 3: Eliminate one variable
We will eliminate \( q \) by multiplying the first equation by \( -3 \) to make the coefficients of \( q \) opposites:
\[ -3(p + q) = -3(40) \]
This simplifies to:
\[ -3p - 3q = -120 \]

Now, our modified system of equations is:
1. \( -3p - 3q = -120 \)
2. \( 7p + 3q = 180 \)

### Step 4: Add the two equations
Add the two equations together to eliminate \( q \):
\[
(-3p - 3q) + (7p + 3q) = -120 + 180
\]
\[
4p = 60
\]
\[
p = 15
\]

### Step 5: Substitute \( p = 15 \) into the first equation
Substitute \( p = 15 \) into the equation \( p + q = 40 \):
\[
15 + q = 40
\]
\[
q = 25
\]

### Conclusion
The values of \( p \) and \( q \) are:
\[ p = 15, \quad q = 25 \]

The correct answer is:
**C) \( p = 15, q = 25 \)**

Would you like more details on the steps or have any questions?

Here are some related questions:
1. What is the elimination method and why is it useful in solving systems of equations?
2. How would you solve this system using the substitution method instead?
3. What are the conditions under which the elimination method is most effective?
4. Could this system have been solved graphically, and what would that look like?
5. What would happen if the equations represented parallel lines?

**Tip:** Always double-check your final values by substituting them back into the original equations to ensure they satisfy both conditions.

Solve the following equations using the elimination method:

p + q - 40 = 0
7p + 3q = 180

What are the values of p and q?
A) p = 20, q = 20
B) p = 30, q = 10
C) p = 15, q = 25
D) p = 10, q = 60

Learn how to solve a system of linear equations using the elimination method. This step-by-step guide shows how to solve p + q - 40 = 0 and 7p + 3q = 180 to find the values of p and q. Suitable for students in Grades 9-12.

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