To solve the system of equations:
[tex]\[ \begin{cases}
3x - 4y = -6 \\
2x + 4y = 16
\end{cases} \][/tex]
we can use the method of elimination to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step-by-Step Solution
1. Add the equations to eliminate [tex]\( y \)[/tex]:
[tex]\[ 3x - 4y = -6 \][/tex]
[tex]\[ 2x + 4y = 16 \][/tex]
If we add these two equations together, the [tex]\( y \)[/tex]-terms will cancel out:
[tex]\[
(3x - 4y) + (2x + 4y) = -6 + 16 \\
3x + 2x - 4y + 4y = -6 + 16 \\
5x = 10
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
5x = 10 \\
x = \frac{10}{5} \\
x = 2
\][/tex]
3. Substitute [tex]\( x = 2 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]:
Let's use the second equation:
[tex]\[ 2x + 4y = 16 \][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
2(2) + 4y = 16 \\
4 + 4y = 16
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
4 + 4y = 16 \\
4y = 16 - 4 \\
4y = 12 \\
y = \frac{12}{4} \\
y = 3
\][/tex]
### Solution
The solution to the system of equations is:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 3 \][/tex]
Thus, the solution is [tex]\((2, 3)\)[/tex].
