To eliminate one of the variables when adding the equations A and B, we can manipulate the coefficients of either \( x \) or \( y \).
Let's start with the two equations:
- \( 6x + 3y = 14 \) (Equation A)
- \( 3x + 2y = 10 \) (Equation B)
To eliminate the variable \( y \), we can make the coefficients of \( y \) in both equations equal. The coefficient of \( y \) in Equation A is 3, and in Equation B, it is 2.
To make these coefficients equal, we can multiply the second equation (Equation B) by 3:
\[ 3(3x + 2y) = 3(10) \]
This results in:
\[ 9x + 6y = 30 \quad \text{(Modified Equation B)} \]
Now, we also need to modify Equation A to introduce a coefficient of 6 for \( y \):
Equation A remains: \[ 6x + 3y = 14 \]
Now, we can rewrite the two equations for addition:
- \( 6x + 3y = 14 \)
- \( 9x + 6y = 30 \)
Now, we can eliminate \( y \). To do this, we can multiply the first equation by 2:
\[ 2(6x + 3y) = 2(14) \] This gives us: \[ 12x + 6y = 28 \quad \text{(Modified Equation A)} \]
Now we can add the modified Equation A and modified Equation B:
- \( 12x + 6y = 28 \)
- \( 9x + 6y = 30 \)
At this point, since both equations now have \( 6y \), when we subtract one from the other, the \( 6y \) terms will cancel out.
Thus, to eliminate \( y \) when Tatum adds the two equations, she should multiply the second equation by \( 3 \).
Answer: 3