To determine if $(5, 6)$ is a solution to the system of equations:

$y = x + 9$ $y = -\frac{1}{5}x + 7,$

we need to check if $(x, y) = (5, 6)$ satisfies both equations.

Step 1: Test the first equation $y = x + 9$

Substitute $x = 5$ and $y = 6$ into $y = x + 9$:

$6 = 5 + 9$ $6 = 14 \quad \text{(False)}.$

This shows that $(5, 6)$ does not satisfy the first equation.

Step 2: Test the second equation $y = -\frac{1}{5}x + 7$

Substitute $x = 5$ and $y = 6$ into $y = -\frac{1}{5}x + 7$:

$6 = -\frac{1}{5}(5) + 7$ $6 = -1 + 7$ $6 = 6 \quad \text{(True)}.$

The point $(5, 6)$ satisfies the second equation.

Final Conclusion:

Since $(5, 6)$ does not satisfy the first equation but does satisfy the second, it is not a solution to the system of equations. The answer is:

$\boxed{\text{No}}$

Would you like further clarification or details on any of the steps?

Related Questions:

1. What are the conditions for a point to be a solution to a system of equations?
2. How do you solve a system of linear equations graphically?
3. Can a point satisfy one equation in a system but not the other? Explain why.
4. What methods can be used to solve systems of equations (e.g., substitution, elimination)?
5. How does the slope of a line affect the solution to a system?

Tip:

Always verify a solution by substituting into all equations of the system!