To solve the system of equations:

1. $2x + 3y = 12$
2. $4x - y = 5$

We will use the substitution or elimination method. Here, we'll proceed with the elimination method.

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Step 1: Eliminate one variable

We first align the equations for elimination. Multiply the second equation by 3 to make the coefficients of $y$ equal:

1. $2x + 3y = 12$
2. $12x - 3y = 15$ (Multiplying $4x - y = 5$ by 3)

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Step 2: Add the equations

Adding the two equations cancels $y$:

$(2x + 3y) + (12x - 3y) = 12 + 15$

$14x = 27$

$x = \frac{27}{14}$

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Step 3: Solve for $y$

Substitute $x = \frac{27}{14}$ into one of the original equations, such as $4x - y = 5$:

$4 \left( \frac{27}{14} \right) - y = 5$

$\frac{108}{14} - y = 5$

$\frac{54}{7} - y = 5$

$y = \frac{54}{7} - 5 = \frac{54}{7} - \frac{35}{7} = \frac{19}{7}$

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Final Solution:

$x = \frac{27}{14}, \, y = \frac{19}{7}$

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Would you like a detailed explanation of any step? Here are some related questions:

1. How can you use substitution instead of elimination to solve this system?
2. Can you verify the solution by substituting $x$ and $y$ back into the original equations?
3. How can graphing be used to solve this system of equations?
4. What does the solution represent geometrically?
5. Can this method handle systems with no solution or infinitely many solutions?

Tip: Always double-check your solutions by substituting them back into the original equations!