Let's solve the given system of linear equations step-by-step:
[tex]\[
\begin{cases}
3x - 4y = -6 \quad (1) \\
2x + 4y = 16 \quad (2)
\end{cases}
\][/tex]
Step 1: Add the two equations to eliminate [tex]\( y \)[/tex].
First, let's add equation (1) to equation (2):
[tex]\[
(3x - 4y) + (2x + 4y) = -6 + 16
\][/tex]
Simplify:
[tex]\[
3x + 2x - 4y + 4y = -6 + 16
\][/tex]
Combine like terms:
[tex]\[
5x = 10
\][/tex]
Step 2: Solve for [tex]\( x \)[/tex].
To find [tex]\( x \)[/tex], divide both sides of the equation by 5:
[tex]\[
x = \frac{10}{5}
\][/tex]
[tex]\[
x = 2
\][/tex]
Step 3: Substitute [tex]\( x = 2 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Let’s use equation (2):
[tex]\[
2x + 4y = 16
\][/tex]
Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[
2(2) + 4y = 16
\][/tex]
Simplify:
[tex]\[
4 + 4y = 16
\][/tex]
Step 4: Isolate [tex]\( y \)[/tex].
Subtract 4 from both sides of the equation:
[tex]\[
4y = 12
\][/tex]
Divide both sides by 4:
[tex]\[
y = \frac{12}{4}
\][/tex]
[tex]\[
y = 3
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
x = 2, \quad y = 3
\][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex].
