Solve \(2x + 3y = 11, 4x - 2y = 10 \)

To solve this system by elimination, we can multiply one or both equations by appropriate constants to make the coefficients of one of the variables the same (but with opposite signs). Let's eliminate \(y\) by multiplying the first equation by 2 and the second equation by 3:

\(
4x + 6y = 22 \\
12x - 6y = 30
\)

Now, we can add the equations:

\(
16x = 52
\)

\(
x = \frac{52}{16} = \frac{13}{4}
\)

Now that we have the value of \(x\), we can substitute it back into one of the original equations to find \(y\). Let's use the first equation:

\(
2\left(\frac{13}{4}\right) + 3y = 11
\)
\(
\frac{13}{2} + 3y = 11
\)
\(
3y = 11 - \frac{13}{2}
\)
\(
3y = \frac{22}{2} - \frac{13}{2} = \frac{9}{2}
\)
\(
y = \frac{9}{6} = \frac{3}{2}
\)

So, the solution to the system is \( x = \frac{13}{4} \) and \( y = \frac{3}{2} \).