To find the equation of the line that passes through $(2, 5)$ and is parallel to the line $x - 2y = 6$, follow these steps:

Step 1: Rewrite the given line in slope-intercept form

The general form of the given line is: $x - 2y = 6$ Solve for $y$ to find the slope: $-2y = -x + 6$ $y = \frac{1}{2}x - 3$ The slope of the given line is $\frac{1}{2}$.

Step 2: Use the point-slope form of a line

Lines that are parallel have the same slope. The equation of the required line will also have a slope of $\frac{1}{2}$. Use the point-slope form: $y - y_1 = m(x - x_1)$ Here, $m = \frac{1}{2}$, $(x_1, y_1) = (2, 5)$: $y - 5 = \frac{1}{2}(x - 2)$

Step 3: Simplify to slope-intercept form

Distribute $\frac{1}{2}$: $y - 5 = \frac{1}{2}x - 1$ Add 5 to both sides: $y = \frac{1}{2}x + 4$

Final Answer:

The equation of the line is: $y = \frac{1}{2}x + 4$

Would you like a graphical representation or further explanation?

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5 Related Questions:

1. How do you find the equation of a line perpendicular to $x - 2y = 6$?
2. What is the significance of the slope in determining parallel lines?
3. Can you derive the equation of a line in standard form instead of slope-intercept form?
4. How would the equation change if the line passed through a different point?
5. What is the general method to convert between different forms of linear equations?

Tip:

Always ensure that two lines are parallel by verifying their slopes are equal.