The line passing through the point (-3, 1) and the point (x, 5) is parallel to the line passing through the points (-2, -1) and (6, 3). What is the value of x?

Introduction / Context:
This is a coordinate geometry question about slopes of lines. Two lines are parallel if and only if they have the same slope. You are given two points on one line and one known point plus one variable x coordinate on the other line. By equating slopes, you can solve for x. This type of question is common in algebra and analytic geometry sections of aptitude and entrance exams.

Given Data / Assumptions:

• First line passes through points A(-3, 1) and B(x, 5).
• Second line passes through points C(-2, -1) and D(6, 3).
• The two lines are parallel.
• We assume standard Cartesian coordinates and the usual definition of slope.

Concept / Approach:
The slope m of a line through points (x1, y1) and (x2, y2) is:

• m = (y2 - y1) / (x2 - x1).

For two lines to be parallel:

• Their slopes must be equal, provided both are non-vertical.
• So slope of AB = slope of CD.

We will compute the slope of CD first since both its coordinates are fully known, and then equate it to the slope of AB to solve for x.

Step-by-Step Solution:
Step 1: Compute slope of line CD through points C(-2, -1) and D(6, 3). Step 2: Slope m_CD = (3 - (-1)) / (6 - (-2)) = (3 + 1) / (6 + 2) = 4 / 8 = 1 / 2. Step 3: Compute slope of line AB through points A(-3, 1) and B(x, 5). Step 4: Slope m_AB = (5 - 1) / (x - (-3)) = 4 / (x + 3). Step 5: Since the lines are parallel, set slopes equal: 4 / (x + 3) = 1 / 2. Step 6: Cross multiply: 4 * 2 = 1 * (x + 3). Step 7: So 8 = x + 3. Step 8: Solve for x: x = 8 - 3 = 5.

Verification / Alternative check:
Now that we have x = 5, recompute the slope of AB to confirm:

• Point B becomes (5, 5).
• Slope m_AB = (5 - 1) / (5 - (-3)) = 4 / 8 = 1 / 2.
• This equals the slope of CD, which we already found to be 1 / 2.

Since both slopes are identical and finite, the two lines are indeed parallel for x = 5. No other value of x satisfies this equality, so the solution is unique.

Why Other Options Are Wrong:
If x = -5, slope m_AB = 4 / (-5 + 3) = 4 / (-2) = -2, which does not equal 1 / 2.
If x = -2, slope m_AB = 4 / (1) = 4, again not equal to 1 / 2.
If x = 2, slope m_AB = 4 / (5) = 0.8, still not equal to 1 / 2.
Therefore, only x = 5 makes the slopes equal and the lines parallel.

Common Pitfalls:
Students sometimes reverse the order of subtraction in the slope formula and thereby introduce sign errors. Another common mistake is to confuse the condition for perpendicular lines (product of slopes equal to -1) with the condition for parallel lines (slopes equal). It is also important to handle negative coordinates carefully when adding or subtracting. Always rewrite expressions like x - (-3) as x + 3 explicitly to avoid sign mistakes.

Final Answer:
The value of x that makes the two lines parallel is 5.