To solve the system of equations given by:
- \( 4y - 12 = 10x \)
- \( 2y - 14 = 4x \)
Let's first rearrange each equation into a more standard form.
From the first equation:
\[ 4y - 10x = 12 \]
From the second equation:
\[ 2y - 4x = 14 \]
Now, we can simplify the second equation. Dividing the entire equation by 2 gives: \[ y - 2x = 7 \] or \[ y = 2x + 7 \]
Now we can substitute this expression for \( y \) into the first equation:
Substituting \( y = 2x + 7 \) into the first equation: \[ 4(2x + 7) - 10x = 12 \]
Expanding: \[ 8x + 28 - 10x = 12 \]
Combining like terms: \[ -2x + 28 = 12 \]
Now, isolate \( x \): \[ -2x = 12 - 28 \] \[ -2x = -16 \] \[ x = 8 \]
Now, substitute \( x = 8 \) back into \( y = 2x + 7 \) to find \( y \): \[ y = 2(8) + 7 = 16 + 7 = 23 \]
Thus, the solution to the system of equations is: \[ \boxed{(8, 23)} \]