In coordinate geometry, find the equation of the line with slope -1/2 that passes through the intersection point of the two lines: x - y = -1 and 3x - 2y = 0.

Introduction / Context:
This coordinate geometry problem combines solving a system of linear equations to find their intersection point with writing the equation of a new line through that point with a specified slope. Such composite questions strengthen your understanding of line equations, intersections, and the slope concept, all of which are fundamental topics in analytic geometry and aptitude tests.

Given Data / Assumptions:

• Lines: x − y = −1 and 3x − 2y = 0.
• The required line must pass through their intersection.
• The slope of the required line is −1/2.
• We must find its equation in standard form.

Concept / Approach:
First, solve the system of equations x − y = −1 and 3x − 2y = 0 to find their intersection point. Then, use the point-slope form of the equation of a line with the given slope and passing through that point. Finally, convert the resulting equation into standard form and compare with the answer choices. This two-step approach ensures both the correct point and correct slope are accounted for.

Step-by-Step Solution:
1) Solve x − y = −1 for y: y = x + 1. 2) Substitute y = x + 1 into 3x − 2y = 0: 3x − 2(x + 1) = 0. 3) Expand: 3x − 2x − 2 = 0 → x − 2 = 0 → x = 2. 4) Substitute x = 2 into y = x + 1: y = 2 + 1 = 3. 5) The intersection point is (2, 3). 6) The required line has slope m = −1/2 and passes through (2, 3). 7) Use point-slope form: y − 3 = (−1/2)(x − 2). 8) Multiply both sides by 2: 2(y − 3) = −(x − 2). 9) Expand: 2y − 6 = −x + 2. 10) Rearrange to standard form: x + 2y − 8 = 0 or x + 2y = 8.

Verification / Alternative check:
Check that the point (2, 3) satisfies x + 2y = 8. Substituting, 2 + 2 * 3 = 2 + 6 = 8, which is correct. The slope of x + 2y = 8 can be found by writing it as y = −(1/2)x + 4, which clearly has slope −1/2. Both the point and the slope match the problem requirements, confirming that x + 2y = 8 is the correct equation.

Why Other Options Are Wrong:
Option b, 3x + y = 7, rearranges to y = −3x + 7, which has slope −3, not −1/2. Option c, x + 2y = −8, and option d, 3x + y = −7, both have incorrect intercepts and do not pass through (2, 3). Option e, y = −x/2, passes through the origin and has the correct slope but does not pass through the intersection point (2, 3). Only option a, x + 2y = 8, satisfies both conditions.

Common Pitfalls:
Errors commonly occur in solving the original system of equations, such as mis substituting y or algebraic sign mistakes. Another frequent issue is mis rearranging the point-slope form to standard form or misreading the slope from the final equation. Carefully solving for the intersection and double-checking both slope and point inclusion will prevent these errors.

Final Answer:
The equation of the line is x + 2y = 8.