To solve the system of equations:

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

Step 1: Solve one equation for one variable

From the first equation: $x = 12 - 2y$

Step 2: Substitute $x = 12 - 2y$ into the second equation

Substitute into $3x - 5y = 3$: $3(12 - 2y) - 5y = 3$ Simplify: $36 - 6y - 5y = 3$ $36 - 11y = 3$ $-11y = 3 - 36$ $-11y = -33$ $y = 3$

Step 3: Substitute $y = 3$ back into $x = 12 - 2y$

$x = 12 - 2(3)$ $x = 12 - 6$ $x = 6$

Final Solution:

$x = 6, \, y = 3$

Verification:

1. Substitute $x = 6, y = 3$ into $x + 2y = 12$: $6 + 2(3) = 12 \quad \text{(True)}$
2. Substitute $x = 6, y = 3$ into $3x - 5y = 3$: $3(6) - 5(3) = 3 \quad \text{(True)}$

The solution satisfies both equations.

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Would you like additional details or explanations?

Here are 5 related questions:

1. How would this method change if the equations were non-linear?
2. Can this system be solved graphically, and how?
3. What happens if the system has no solution or infinitely many solutions?
4. How can substitution be compared to elimination in solving such systems?
5. How would this be solved using matrix methods?

Tip: Always verify your solution by substituting the values back into the original equations.