In coordinate geometry, the line segment AB has slope 4/3. The coordinates of point A are (x, -5) and the coordinates of point B are (2, -3). What is the value of x?

Introduction / Context:
This question tests a fundamental concept from coordinate geometry, namely the slope of a line segment joining two points. You are given the slope value and the coordinates of the two endpoints in partially unknown form, and you need to find the missing x coordinate of one point using the slope formula.

Given Data / Assumptions:
• Slope of line segment AB is 4/3.
• Coordinates of A are (x, -5).
• Coordinates of B are (2, -3).
• The line through A and B is non vertical so the usual slope formula is valid.

Concept / Approach:
The slope m of the line joining two points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1). Here the slope is already known and the unknown is x. By substituting the given coordinates into the formula, you get a simple linear equation in x which can be solved using basic algebra.

Step-by-Step Solution:
Step 1: Write the slope formula for points A(x, -5) and B(2, -3): m = (y2 - y1) / (x2 - x1) = [(-3) - (-5)] / [2 - x]. Step 2: Simplify the numerator: -3 - (-5) = -3 + 5 = 2. So slope m = 2 / (2 - x). Step 3: The given slope is 4/3, so set 2 / (2 - x) = 4/3. Step 4: Cross multiply: 2 * 3 = 4 * (2 - x). Step 5: This gives 6 = 8 - 4x. Step 6: Rearrange to solve for x: 4x = 8 - 6 = 2, so x = 2 / 4 = 1/2.

Verification / Alternative check:
Substitute x = 1/2 back into the slope expression. Then the denominator 2 - x becomes 2 - 1/2 = 3/2. So the slope is 2 / (3/2) = 2 * (2/3) = 4/3, which matches the given slope. This confirms that x = 1/2 is correct.

Why Other Options Are Wrong:
• -1/4 makes the denominator 2 - (-1/4) = 9/4, giving slope 2 / (9/4) = 8/9, not 4/3.
• 4 makes the denominator 2 - 4 = -2, giving slope 2 / (-2) = -1, not 4/3.
• -4 makes the denominator 2 - (-4) = 6, giving slope 2 / 6 = 1/3, not 4/3.

Common Pitfalls:
Many learners forget the correct order in the slope formula and sometimes interchange x1 and x2 or y1 and y2 inconsistently. As long as both numerator and denominator use the same order for points, the slope remains correct. Another common mistake is sign handling when subtracting negative numbers. Carefully simplifying -3 - (-5) as -3 + 5 is essential.

Final Answer:
The value of x is 1/2.