To determine if the value of variable \( a \) can be found using the given midpoint \( M \) of line segment \( FG \), we begin by using the midpoint formula. The midpoint \( M \) of a segment with endpoints \( F(x_1, y_1) \) and \( G(x_2, y_2) \) is calculated as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Given:
- \( F(b + 1, a + 2) \)
- \( M(3, 5) \)
- \( G(2a, 3b + 3) \)
From the midpoint \( M \) coordinates, we can set up two equations based on the \( x \)-coordinates and \( y \)-coordinates.
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For the \( x \)-coordinates: \[ \frac{(b + 1) + (2a)}{2} = 3 \]
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For the \( y \)-coordinates: \[ \frac{(a + 2) + (3b + 3)}{2} = 5 \]
Now, let's solve these equations:
For the \( x \)-coordinates:
Multiply both sides by \( 2 \): \[ (b + 1) + 2a = 6 \quad \Rightarrow \quad b + 2a = 5 \quad \text{(Equation 1)} \]
For the \( y \)-coordinates:
Multiply both sides by \( 2 \): \[ (a + 2) + (3b + 3) = 10 \quad \Rightarrow \quad a + 3b + 5 = 10 \quad \Rightarrow \quad a + 3b = 5 \quad \text{(Equation 2)} \]
We now have a system of equations:
- \( b + 2a = 5 \)
- \( a + 3b = 5 \)
To solve this system for \( a \) and \( b \), we can express \( b \) from Equation 1: \[ b = 5 - 2a \]
Now substitute \( b \) into Equation 2: \[ a + 3(5 - 2a) = 5 \ a + 15 - 6a = 5 \ -5a + 15 = 5 \ -5a = 5 - 15 \ -5a = -10 \ a = 2 \]
Since we found \( a = 2 \), we have determined a specific value for \( a \).
Conclusion:
So the correct choice is:
A. To determine the value of \( a \), set up and solve a system of equations in two variables. The value of \( a \) is 2.