To solve the given system of equations:

$x - y = 0$ $x + y = 6$

we can use substitution or elimination. Here, substitution will work well:

1. Solve for $x$ in terms of $y$ from the first equation: $x - y = 0 \implies x = y$

2. Substitute $x = y$ into the second equation: $x + y = 6 \implies y + y = 6 \implies 2y = 6 \implies y = 3$

3. Substitute $y = 3$ back into $x = y$: $x = 3$

Thus, the solution to the system is: $(x, y) = (3, 3)$

Verification

To verify graphically, we can plot the two equations:

1. $x - y = 0$ (or $y = x$), which is a line through the origin with a slope of 1.
2. $x + y = 6$ (or $y = 6 - x$), which is a line with a slope of -1 that intersects the y-axis at 6.

These two lines intersect at the point $(3, 3)$, confirming that our solution is correct.

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Would you like to see the graphical verification or need further details on this solution?

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Related Questions:

1. How would the solution change if the second equation was $x + y = 4$ instead of $x + y = 6$?
2. What would happen if the equations were $x - y = 0$ and $2x + 2y = 12$?
3. How can you determine if two linear equations will have exactly one solution, no solution, or infinitely many solutions?
4. How would the system be solved if one equation were quadratic, e.g., $x^2 - y = 0$?
5. How do you determine the slope and intercept of each line in a system of linear equations?

Tip: A system of two linear equations in two variables is consistent and independent if the lines intersect at exactly one point, yielding a unique solution.