If `2p+q=11 and p+2q=13,` then p+q=

Options

A. `6`
B. `8`
C. `9`
D. `12`

Answer

To solve the equations \(2p + q = 11\) and \(p + 2q = 13\) and find the value of \(p + q\), we can follow these steps:

### Step 1: Write down the equations
We have:
1. \(2p + q = 11\) (Equation 1)
2. \(p + 2q = 13\) (Equation 2)

### Step 2: Multiply Equation 1 by 2
To eliminate \(q\), we can multiply Equation 1 by 2:
\[ 2(2p + q) = 2(11) \]
This gives us:
\[ 4p + 2q = 22 \quad (Equation 3) \]

### Step 3: Subtract Equation 2 from Equation 3
Now we will subtract Equation 2 from Equation 3:
\[ (4p + 2q) - (p + 2q) = 22 - 13 \]
This simplifies to:
\[ 4p + 2q - p - 2q = 9 \]
So we have:
\[ 3p = 9 \]

### Step 4: Solve for \(p\)
Now, divide both sides by 3:
\[ p = \frac{9}{3} = 3 \]

### Step 5: Substitute \(p\) back into one of the original equations
We can substitute \(p = 3\) into Equation 1:
\[ 2(3) + q = 11 \]
This simplifies to:
\[ 6 + q = 11 \]
Now, solve for \(q\):
\[ q = 11 - 6 = 5 \]

### Step 6: Find \(p + q\)
Now that we have \(p\) and \(q\):
\[ p + q = 3 + 5 = 8 \]

### Final Answer
Thus, the value of \(p + q\) is \(8\).

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