To eliminate the [tex]\( y \)[/tex]-terms using the linear combination method, we need to multiply the second equation in such a way that the coefficients of [tex]\( y \)[/tex] in both equations become opposites (i.e., they sum to zero).
We start with the given system of equations:
[tex]\[
\begin{array}{r}
7x + 16y = -2 \tag{1} \\
3x - 4y = 22 \tag{2}
\end{array}
\][/tex]
To eliminate [tex]\( y \)[/tex], we need to make the coefficient of [tex]\( y \)[/tex] in the second equation the negative of the coefficient of [tex]\( y \)[/tex] in the first equation. The coefficient of [tex]\( y \)[/tex] in the first equation is [tex]\( 16 \)[/tex].
The coefficient of [tex]\( y \)[/tex] in the second equation is [tex]\( -4 \)[/tex]. To determine by what number should we multiply the second equation so that the coefficient of [tex]\( y \)[/tex] in the second equation becomes [tex]\( -16 \)[/tex] (the negative of the coefficient of [tex]\( y \)[/tex] in the first equation), we need to solve for the multiplier:
[tex]\[
\text{multiplier} \times (-4) = -16
\][/tex]
Solving for the multiplier, we divide both sides of the equation by [tex]\(-4\)[/tex]:
[tex]\[
\text{multiplier} = \frac{-16}{-4} = 4
\][/tex]
Thus, the second equation should be multiplied by [tex]\( 4 \)[/tex] to eliminate the [tex]\( y \)[/tex]-term by the linear combination method.
So the correct answer is:
[tex]\[ 4 \][/tex]
