The slope between two points equals the change in y divided by the change in x.\nThe slope of line segment AB is -2/3. If A = (x, -3) and B = (5, 2), what is the value of x?

Introduction / Context:
This problem tests coordinate geometry basics: computing slope from two points and solving for an unknown coordinate. The slope formula is m = (y2 - y1)/(x2 - x1). Here, one x-coordinate is unknown, and we use the given slope to form an equation and solve it.

Given Data / Assumptions:

• Slope m = -2/3
• Point A = (x, -3)
• Point B = (5, 2)
• Use slope formula m = (y_B - y_A)/(x_B - x_A)

Concept / Approach:
Substitute coordinates into the slope formula. Then solve the resulting fraction equation by cross-multiplication. Carefully keep the order of subtraction consistent because changing point order changes the sign of both numerator and denominator, but the slope must remain the given value.

Step-by-Step Solution:
Slope between A(x, -3) and B(5, 2): m = (2 - (-3)) / (5 - x) Compute numerator: 2 - (-3) = 5 So m = 5 / (5 - x) Given m = -2/3, set 5 / (5 - x) = -2/3 Cross-multiply: 5*3 = -2*(5 - x) 15 = -10 + 2x Add 10 to both sides: 25 = 2x So x = 25/2 = 12.5

Verification / Alternative check:
Plug x = 12.5 back: denominator 5 - 12.5 = -7.5, so slope = 5/(-7.5) = -2/3, correct. This confirms the solution.

Why Other Options Are Wrong:
10 and 4 typically come from using (x - 5) instead of (5 - x) without adjusting signs properly.
-4 and -14 come from sign errors when distributing -2 across (5 - x).

Common Pitfalls:
Forgetting that 5 - x can be negative, mishandling negative signs during cross-multiplication, or incorrectly computing 2 - (-3). Another common mistake is swapping the denominator to x - 5 but not changing the sign of the slope.

Final Answer:
x = 12.5