Solve the linear equation 3x − 8(2 − x) = −19 and find the value of x.

Introduction / Context:
This question is a straightforward algebraic problem involving expansion of brackets and solving a linear equation in one variable. These steps form the foundation for more advanced algebra topics and appear frequently in quantitative aptitude tests, where speed and accuracy are both important.

Given Data / Assumptions:
- The equation is 3x − 8(2 − x) = −19.
- x is a real number.
- We must isolate x and find its numerical value.

Concept / Approach:
The procedure is to expand the bracket term using the distributive property, collect like terms on one side, and then perform inverse operations to solve for x. As always, checking the solution by substitution confirms that no algebraic error has been made.

Step-by-Step Solution:
Step 1: Start with the given equation 3x − 8(2 − x) = −19. Step 2: Distribute the −8 across the bracket. Multiply −8 by 2 to get −16, and −8 by −x to get +8x. Thus 3x − 8(2 − x) becomes 3x − 16 + 8x. Step 3: Combine like terms on the left hand side: 3x + 8x = 11x, so the equation becomes 11x − 16 = −19. Step 4: Add 16 to both sides to move the constant term: 11x = −19 + 16 = −3. Step 5: Divide both sides by 11 to solve for x: x = −3 / 11. Step 6: Therefore, the solution of the equation is x = −3/11.
Verification / Alternative check:
Substitute x = −3/11 back into the original equation. Compute 3x = 3 * (−3/11) = −9/11. Next, 2 − x = 2 − (−3/11) = 2 + 3/11 = 25/11. Then 8(2 − x) = 8 * 25/11 = 200/11. The left hand side becomes 3x − 8(2 − x) = (−9/11) − (200/11) = −209/11, which simplifies to −19. This matches the right hand side, confirming that x = −3/11 is correct.

Why Other Options Are Wrong:
Option b, −33/11, simplifies to −3, and substituting x = −3 yields 3(−3) − 8(2 − (−3)) = −9 − 8(5) = −9 − 40 = −49, not −19.
Option c, −3/5, and option d, −33/5, when substituted, also fail to satisfy the equation; they give different left hand side values from −19.
Option e, 0, produces 3(0) − 8(2 − 0) = −16, which again does not match the required right hand side −19.

Common Pitfalls:
Students sometimes forget to distribute the negative sign correctly when expanding −8(2 − x), leading to an incorrect sign for the term with x. Another typical mistake is mishandling the arithmetic when combining constants or dividing by the coefficient of x. Writing each step explicitly and performing a substitution check at the end is an effective way to avoid such mistakes and to build confidence in solving linear equations.

Final Answer:
The solution of the equation is x = −3/11.