Which of the following can be a solution to the pair of equations `2q-rt=21` and `q+rt=36` ?

Options

A. q = 18 and t = 2
B. q = 19 and t = 1
C. q = 20 and t = -1
D. q = 21 and t = -2

Explanation

To solve the pair of equations \(2q - rt = 21\) and \(q + rt = 36\), we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(2q - rt = 21\) (Equation 1)
2. \(q + rt = 36\) (Equation 2)

### Step 2: Add the two equations
To eliminate \(rt\), we can add Equation 1 and Equation 2 together:
\[ (2q - rt) + (q + rt) = 21 + 36 \]

### Step 3: Simplify the left side
On the left side, \( -rt + rt\) cancels out:
\[ 2q + q = 3q \]
Thus, we have:
\[ 3q = 57 \]

### Step 4: Solve for \(q\)
Now, divide both sides by 3 to find \(q\):
\[ q = \frac{57}{3} = 19 \]

### Step 5: Substitute \(q\) back to find \(rt\)
Now that we have \(q = 19\), we can substitute this value back into either of the original equations to find \(rt\). Let's use Equation 2:
\[ q + rt = 36 \]
Substituting \(q = 19\):
\[ 19 + rt = 36 \]
Now, solve for \(rt\):
\[ rt = 36 - 19 = 17 \]

### Conclusion
We have found \(q = 19\) and \(rt = 17\). Now, we need to check which of the options provided has \(q = 19\).