We are given the system of linear equations:

1. $x + 6y = 2$
2. $x - 5y = -9$

Step 1: Solve one equation for $x$

From the first equation: $x = 2 - 6y$

Step 2: Substitute $x$ into the second equation

Substitute $x = 2 - 6y$ into the second equation $x - 5y = -9$: $(2 - 6y) - 5y = -9$ Simplify: $2 - 6y - 5y = -9$ $2 - 11y = -9$ $-11y = -11$ $y = 1$

Step 3: Solve for $x$

Substitute $y = 1$ into $x = 2 - 6y$: $x = 2 - 6(1)$ $x = 2 - 6$ $x = -4$

Final Answer:

$x = -4, \, y = 1$

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Verification:

Substitute $x = -4$ and $y = 1$ into both equations to verify:

1. $x + 6y = 2$:
   $(-4) + 6(1) = 2 \implies 2 = 2$ ✅

2. $x - 5y = -9$:
   $(-4) - 5(1) = -9 \implies -9 = -9$ ✅

The solution is correct.

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Do you want me to provide a graphical explanation or explore any specific steps?

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Related Questions:

1. What is the geometric interpretation of the solution to this system of equations?
2. How would the solution change if the equations were $x + 6y = 5$ and $x - 5y = -7$?
3. Can this system be solved using the substitution method as we did here?
4. What is the determinant of the coefficient matrix of this system?
5. How can you solve this system using matrices or Cramer's Rule?

Tip:

For systems of linear equations, substitution is effective when one variable can be easily isolated, while elimination is more efficient for simplifying simultaneous equations.