Solve the System: Using Substitution in 3x - 2y and 4x + 3y Equations

To solve the given system of equations using substitution, follow these steps:

First, we simplify the given equations. Let's start with the first equation:

$\frac{3x-2y}{8} + \frac{x-y}{2} = 14.$

Find a common denominator for fractions on the left side. The common denominator of 8 and 2 is 8:

$\frac{3x-2y}{8} + \frac{4(x-y)}{8} = 14 \rightarrow \frac{3x-2y+4x-4y}{8} = 14.$

Simplify inside the fraction:

$\frac{7x-6y}{8} = 14 \rightarrow 7x - 6y = 112. \quad \text{(multiply through by 8)}$

This is our simplified form for the first equation.

Now, simplify the second equation:

$\frac{4x+3y}{5} - \frac{2x-2y}{3} = 20.$

Find a common denominator for the fractions, which is 15:

$\frac{3(4x + 3y) - 5(2x - 2y)}{15} = 20.$

Distribute and simplify:

$\frac{12x + 9y - 10x + 10y}{15} = 20 \rightarrow \frac{2x + 19y}{15} = 20.$

Multiply through by 15 to clear the fraction:

$2x + 19y = 300.$

Now, we have two simplified equations:

$\begin{cases} 7x - 6y = 112 \\ 2x + 19y = 300 \end{cases}.$

From the first equation, solve for $x$:

$7x = 6y + 112 \rightarrow x = \frac{6y + 112}{7}.$

Substitute $x = \frac{6y + 112}{7}$ into the second equation:

$2\left(\frac{6y + 112}{7}\right) + 19y = 300.$

Distribute:

$\frac{12y + 224}{7} + 19y = 300.$

Clear the fraction by multiplying through by 7:

$12y + 224 + 133y = 2100.$

Combine like terms:

$145y + 224 = 2100.$

Subtract 224 from both sides:

$145y = 1876 \rightarrow y = \frac{1876}{145} \approx 12.93.$

Now, substitute $y \approx 12.93$ back into $x = \frac{6y + 112}{7}$:

$x = \frac{6(12.93) + 112}{7} = \frac{77.58 + 112}{7} = \frac{189.58}{7} \approx 27.08.$

Therefore, the solution to the system is:

$x \approx 27.08, \, y \approx 12.93$.

This corresponds to choice 1:

$x = 27.08, y = 12.93$.