2p+3q=18 and `4p^2+4pq–3q2−36=0 `then what is (2p+q) equal to?

Options

A. 6
B. 7
C. 10
D. 20

Explanation

To solve the problem step by step, we will start with the two equations provided:

1. \( 2p + 3q = 18 \)
2. \( 4p^2 + 4pq - 3q^2 - 36 = 0 \)

We need to find the value of \( 2p + q \).

### Step 1: Express \( p \) in terms of \( q \)

From the first equation, we can express \( p \) in terms of \( q \):

\[ 2p = 18 - 3q \]
\[ p = \frac{18 - 3q}{2} \]

### Step 2: Substitute \( p \) into the second equation

Now, we will substitute this expression for \( p \) into the second equation:

\[ 4\left(\frac{18 - 3q}{2}\right)^2 + 4\left(\frac{18 - 3q}{2}\right)q - 3q^2 - 36 = 0 \]

### Step 3: Simplify the equation

Calculating \( 4\left(\frac{18 - 3q}{2}\right)^2 \):

\[ = 4 \cdot \frac{(18 - 3q)^2}{4} = (18 - 3q)^2 \]

Calculating \( 4\left(\frac{18 - 3q}{2}\right)q \):

\[ = 2(18 - 3q)q = 36q - 6q^2 \]

Substituting these back into the equation gives:

\[ (18 - 3q)^2 + (36q - 6q^2) - 3q^2 - 36 = 0 \]

### Step 4: Expand and combine like terms

Expanding \( (18 - 3q)^2 \):

\[ = 324 - 108q + 9q^2 \]

So, the equation becomes:

\[ 324 - 108q + 9q^2 + 36q - 6q^2 - 3q^2 - 36 = 0 \]

Combining like terms:

\[ (9q^2 - 6q^2 - 3q^2) + (-108q + 36q) + (324 - 36) = 0 \]
\[ 0 + (-72q) + 288 = 0 \]

This simplifies to:

\[ -72q + 288 = 0 \]

### Step 5: Solve for \( q \)

Rearranging gives:

\[ 72q = 288 \]
\[ q = \frac{288}{72} = 4 \]

### Step 6: Substitute \( q \) back to find \( p \)

Now substitute \( q = 4 \) back into the equation for \( p \):

\[ 2p + 3(4) = 18 \]
\[ 2p + 12 = 18 \]
\[ 2p = 18 - 12 \]
\[ 2p = 6 \]
\[ p = 3 \]

### Step 7: Find \( 2p + q \)

Now we can find \( 2p + q \):

\[ 2p + q = 2(3) + 4 = 6 + 4 = 10 \]

Thus, the value of \( 2p + q \) is \( 10 \).

### Final Answer

\[ \boxed{10} \]