To solve this problem, we need to find the equation of a line that passes through the point $(2, 6)$ and is parallel to the line given by the equation $x - y = 2$.

Step 1: Determine the Slope of the Given Line

The equation of the given line is: $x - y = 2$ To find the slope, we need to rewrite this in slope-intercept form ($y = mx + b$), where $m$ represents the slope.

Starting with: $x - y = 2$ Subtract $x$ from both sides: $-y = -x + 2$ Then, divide by $-1$ to solve for $y$: $y = x - 2$ So, the slope of this line ($m$) is $1$.

Step 2: Write the Equation of the Parallel Line

Since the line we are looking for is parallel, it will have the same slope, $m = 1$.

Using the point-slope form of a line equation, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point the line passes through, substitute $(2, 6)$ and $m = 1$: $y - 6 = 1(x - 2)$ Expanding this: $y - 6 = x - 2$ Add 6 to both sides: $y = x + 4$

Final Answer

The equation of the line that passes through $(2, 6)$ and is parallel to $x - y = 2$ is: $y = x + 4$

Would you like further details on this, or do you have any questions?

Here are some related questions that might deepen your understanding:

1. How would the equation change if the line were perpendicular instead of parallel?
2. Can you explain why parallel lines have the same slope?
3. How would you find the equation of a line passing through a point but having a different slope?
4. What would be the effect of changing the y-intercept in a parallel line?
5. How can you determine the slope if the equation isn't in standard or slope-intercept form?

Tip: Always rewrite a line equation in slope-intercept form ($y = mx + b$) to identify the slope quickly!