Solve the Equations: 3x-y=5 and 5x+2y=12 for x and y

To solve the system of equations using the elimination method, follow these steps:

• Step 1: Write down the given equations:
  $3x - y = 5$ (Equation 1)
  $5x + 2y = 12$ (Equation 2)
• Step 2: Align coefficients for elimination. Here, we aim to eliminate $y$ by aligning coefficients of $y$ in both equations. Multiply Equation 1 by $2$:
  $2(3x - y) = 2(5)$, which simplifies to $6x - 2y = 10$.
• Step 3: Add the modified Equation 1 to Equation 2:
  $(6x - 2y) + (5x + 2y) = 10 + 12$.
  The $2y$ terms cancel out, leaving $11x = 22$.
• Step 4: Solve for $x$:
  $11x = 22$ simplifies to $x = 2$.
• Step 5: Substitute $x = 2$ back into Equation 1 to solve for $y$:
  $3(2) - y = 5$, which simplifies to $6 - y = 5$. Thus, $y = 1$.

Therefore, the solution to the system of equations is $x = 2$ and $y = 1$.

This corresponds to choice 2 in the provided answer choices.