Let's solve the system of equations step by step:
[tex]\[
\begin{cases}
3x - 2y = 5 \\
2x + 4y = 14
\end{cases}
\][/tex]
### Step 1: Solve one equation for one of the variables
Let's solve the first equation for \( x \):
[tex]\[
3x - 2y = 5
\][/tex]
Add \( 2y \) to both sides:
[tex]\[
3x = 5 + 2y
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{5 + 2y}{3}
\][/tex]
### Step 2: Substitute into the other equation
Substitute \( x = \frac{5 + 2y}{3} \) into the second equation \( 2x + 4y = 14 \):
[tex]\[
2\left(\frac{5 + 2y}{3}\right) + 4y = 14
\][/tex]
Multiply through by 3 to clear the fraction:
[tex]\[
2(5 + 2y) + 12y = 42
\][/tex]
Distribute \( 2 \):
[tex]\[
10 + 4y + 12y = 42
\][/tex]
Combine like terms:
[tex]\[
10 + 16y = 42
\][/tex]
Subtract 10 from both sides:
[tex]\[
16y = 32
\][/tex]
Divide by 16:
[tex]\[
y = 2
\][/tex]
### Step 3: Substitute back to find \( x \)
Substitute \( y = 2 \) back into the equation \( x = \frac{5 + 2y}{3} \):
[tex]\[
x = \frac{5 + 2(2)}{3}
\][/tex]
Simplify:
[tex]\[
x = \frac{5 + 4}{3} = \frac{9}{3} = 3
\][/tex]
### Final Solution
The solution to the system of equations is:
[tex]\[
x = 3, \quad y = 2
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = 3 \)[/tex] and [tex]\( y = 2 \)[/tex].
