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 ($x + 2y = 12$), solve for $x$: $x = 12 - 2y$

Step 2: Substitute into the second equation

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

Step 3: Substitute $y = 3$ back into the first equation

Substitute $y = 3$ into $x + 2y = 12$: $x + 2(3) = 12$ $x + 6 = 12$ $x = 6$

Final Solution:

$x = 6, \, y = 3$

Would you like a step-by-step breakdown or have any specific questions?

Relative Questions:

1. How would the solution change if the coefficients in the equations were altered?
2. What is the geometric interpretation of the solution $(6, 3)$?
3. How can substitution be replaced by elimination in solving this system?
4. What if the system had no solution or infinite solutions? How would we identify that?
5. Can this system be solved using matrix methods? If so, how?

Tip:

Always double-check your solution by substituting the values back into the original equations.