The slope of line segment AB is −4/3. The coordinates of A and B are A(x, −5) and B(−5, 3) respectively. What is the value of x?

Introduction / Context:
This question tests your understanding of the slope formula in coordinate geometry and your ability to use it to find an unknown coordinate. The slope gives the steepness of the line, and knowing two points with an unknown coordinate allows you to set up an equation and solve it for that coordinate.

Given Data / Assumptions:

• Line segment AB has slope m = −4/3.

• Point A has coordinates (x, −5).

• Point B has coordinates (−5, 3).

• We must find the value of x that makes the slope correct.

Concept / Approach:
The slope m of a line passing through points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ − y₁) / (x₂ − x₁). By substituting the coordinates of A and B into this formula and equating the result to the given slope −4/3, we obtain a simple linear equation in x that can be solved easily.

Step-by-Step Solution:
Step 1: Label A(x₁, y₁) = (x, −5) and B(x₂, y₂) = (−5, 3). Step 2: Use the slope formula m = (y₂ − y₁) / (x₂ − x₁). Step 3: Compute the numerator: y₂ − y₁ = 3 − (−5) = 3 + 5 = 8. Step 4: Compute the denominator: x₂ − x₁ = −5 − x. Step 5: So the slope from A to B is 8 / (−5 − x) = −4/3. Step 6: Set up the equation 8 / (−5 − x) = −4/3. Step 7: Cross multiply: 8 * 3 = −4 * (−5 − x). Step 8: This gives 24 = −4(−5 − x) = 20 + 4x. Step 9: Solve for x: 24 − 20 = 4x, so 4 = 4x and x = 1.

Verification / Alternative check:
Substitute x = 1 back into the coordinates of A, giving A(1, −5). Now the slope between A(1, −5) and B(−5, 3) is (3 − (−5)) / (−5 − 1) = 8 / (−6) = −4/3, which matches the given slope. Therefore, x = 1 is confirmed as correct.

Why Other Options Are Wrong:
Option −1: With A(−1, −5), the slope becomes 8 / (−5 − (−1)) = 8 / (−4) = −2, not −4/3.
Option 2: With A(2, −5), the slope is 8 / (−7), which is not equal to −4/3.
Option −2: With A(−2, −5), the slope is 8 / (−3), which is again not equal to −4/3.

Common Pitfalls:
A frequent mistake is reversing the order of points incorrectly, which changes the signs in both numerator and denominator, though here that would still give the same slope. A more serious error is mismanaging the negative signs during cross multiplication, leading to wrong values of x. Writing each step clearly and checking the final slope with the found value of x helps avoid these errors.

Final Answer:
The correct value of x is 1.