To solve for [tex]\( y \)[/tex] in the given system of equations:
[tex]\[
\begin{array}{l}
3x - 4y = 10 \\
2x - 4y = 6
\end{array}
\][/tex]
we will use the method of elimination.
1. First, let's subtract the second equation from the first equation. This will help eliminate [tex]\( y \)[/tex] from the equations:
[tex]\[
(3x - 4y) - (2x - 4y) = 10 - 6
\][/tex]
2. Simplify the left-hand side and the right-hand side of the equation:
[tex]\[
(3x - 2x) - (4y - 4y) = 4
\][/tex]
[tex]\[
x = 4
\][/tex]
3. Now that we have [tex]\( x = 4 \)[/tex], we substitute this value back into one of the original equations to solve for [tex]\( y \)[/tex]. Let's substitute it into the first equation:
[tex]\[
3(4) - 4y = 10
\][/tex]
4. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[
12 - 4y = 10
\][/tex]
[tex]\[
-4y = 10 - 12
\][/tex]
[tex]\[
-4y = -2
\][/tex]
[tex]\[
y = \frac{-2}{-4}
\][/tex]
[tex]\[
y = \frac{1}{2}
\][/tex]
Thus, the value of [tex]\( y \)[/tex] that satisfies the given system of equations is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
