We are solving the system of equations:

1. $x + 7y = 12$
2. $3x - 5y = 10$

Step 1: Express $x$ from the first equation

From $x + 7y = 12$, we solve for $x$: $x = 12 - 7y$

Step 2: Substitute $x$ into the second equation

Substitute $x = 12 - 7y$ into $3x - 5y = 10$: $3(12 - 7y) - 5y = 10$

Expand: $36 - 21y - 5y = 10$

Combine like terms: $36 - 26y = 10$

Simplify: $-26y = 10 - 36$ $-26y = -26$ $y = 1$

Step 3: Solve for $x$

Substitute $y = 1$ into $x = 12 - 7y$: $x = 12 - 7(1)$ $x = 12 - 7$ $x = 5$

Final Solution:

$x = 5, \, y = 1$

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Would you like a step-by-step verification or graph of the solution? Here are related questions for deeper understanding:

1. How do you verify the solution of a system of linear equations?
2. What are other methods to solve such systems (e.g., substitution, elimination)?
3. How do you represent these equations graphically?
4. Can a system of linear equations have no solution or infinite solutions?
5. How is solving a system of equations applied in real-world problems?

Tip: Always check your solution by substituting it back into the original equations!