To eliminate the \( y \) terms from the two equations, we want to make the coefficients of \( y \) in both equations equal in magnitude but opposite in sign, so that when we add them together, the \( y \) terms will cancel out.
The original equations are:
- \( 4x - 3y = 34 \)
- \( 3x + 2y = 17 \)
The coefficients of \( y \) are \(-3\) in the first equation and \(2\) in the second equation. We can make these coefficients equal by multiplying the first equation by \( 2 \) (to make the coefficient of \( y \) equal to \( -6 \)) and the second equation by \( 3 \) (to make the coefficient of \( y \) equal to \( 6 \)).
Here are the equations after multiplying:
- First equation: \( 2(4x - 3y) = 2(34) \) → \( 8x - 6y = 68 \)
- Second equation: \( 3(3x + 2y) = 3(17) \) → \( 9x + 6y = 51 \)
Now, if we sum these two adjusted equations:
\[ (8x - 6y) + (9x + 6y) = 68 + 51 \]
The \( -6y \) and \( +6y \) will cancel out, leaving us with:
\[ 17x = 119 \]
Now, we can solve for \( x \).
Thus, the correct choice for multiplying the equations is:
The first equation should be multiplied by 2 and the second equation by 3.