Solve the linear equation (8 − 10x) − (13x − 2) = −9 and determine the value of x.

Introduction / Context:
This question is a straightforward test of algebraic manipulation and solving a simple linear equation in one variable. Linear equations appear in almost every branch of mathematics and in many aptitude questions, so you must be comfortable expanding brackets, collecting like terms, and isolating the variable to find its value.

Given Data / Assumptions:
- The equation is (8 − 10x) − (13x − 2) = −9.
- x is a real number.
- We need to simplify the left hand side and solve for x.

Concept / Approach:
The approach is to first remove parentheses carefully, especially noting the negative sign before the second bracket. Then combine like terms to obtain a standard linear form ax + b = c. Finally, isolate x using basic algebraic operations. Checking the solution by substitution is a good habit to confirm accuracy.

Step-by-Step Solution:
Step 1: Start with the given equation (8 − 10x) − (13x − 2) = −9. Step 2: Distribute the minus sign across the second bracket: (8 − 10x) − 13x + 2 = −9. Step 3: Combine constant terms and x terms on the left hand side. Constants: 8 + 2 = 10. x terms: −10x − 13x = −23x. Step 4: The equation becomes 10 − 23x = −9. Step 5: Subtract 10 from both sides to move the constant term: −23x = −9 − 10 = −19. Step 6: Divide both sides by −23 to solve for x: x = (−19) / (−23) = 19 / 23.
Verification / Alternative check:
Substitute x = 19 / 23 back into the original equation to verify. Compute 8 − 10x = 8 − 10 * (19 / 23) = 8 − 190 / 23. Convert 8 to 184 / 23, giving (184 − 190) / 23 = −6 / 23. Next, 13x − 2 = 13 * (19 / 23) − 2 = 247 / 23 − 2. Convert 2 to 46 / 23 and obtain (247 − 46) / 23 = 201 / 23. The left hand side becomes (8 − 10x) − (13x − 2) = (−6 / 23) − (201 / 23) = −207 / 23, which simplifies to −9. This matches the right hand side exactly, confirming the solution is correct.

Why Other Options Are Wrong:
Option a, −19/23, would produce a different value on substitution, not equal to −9 on the right hand side.
Options c and d, −1/23 and 1/23, are too small in magnitude and also fail when substituted back.
Option e, 0, leads to (8 − 0) − (0 − 2) = 8 − (−2) = 10, which is not equal to −9.

Common Pitfalls:
Students often make sign errors when removing brackets, especially when a negative sign precedes the bracket. Forgetting to subtract 2 instead of adding it or miscombining terms can lead to an incorrect coefficient of x. Another mistake is to divide by the wrong sign when isolating x. Always distribute negative signs with care and check the solution by substitution to avoid these errors.

Final Answer:
The value of x that satisfies the equation is 19/23.