If the two straight lines p x + y = 2 and 3 x + y = 4 are parallel to each other, what is the value of p?

Introduction / Context:
This arithmetic reasoning question tests your understanding of the condition for two straight lines to be parallel in coordinate geometry. When two linear equations in x and y represent parallel lines, their slopes must be equal. By carefully comparing the coefficients of x and y in the given equations, we can find the required value of p that makes the two lines parallel.

Given Data / Assumptions:
- First line: p x + y = 2.- Second line: 3 x + y = 4.- Both equations are in the standard linear form a x + b y = c.- We assume x and y are real variables and p is a real constant to be determined.

Concept / Approach:
For a straight line written as a x + b y = c, the slope m is given by m = -a / b, provided b is not zero. Two lines are parallel if and only if they have the same slope but different intercepts. Therefore, if line 1 has slope m1 and line 2 has slope m2, the condition for parallelism is m1 = m2. We will compute the slopes of both lines and equate them to solve for p. The constant terms on the right side do not affect the slopes, only the intercepts.

Step-by-Step Solution:
- Rewrite the first line p x + y = 2. Here a1 = p and b1 = 1.- The slope of the first line is m1 = -a1 / b1 = -p / 1 = -p.- Rewrite the second line 3 x + y = 4. Here a2 = 3 and b2 = 1.- The slope of the second line is m2 = -a2 / b2 = -3 / 1 = -3.- For the lines to be parallel, set m1 = m2, so -p = -3.- Multiply both sides by -1 to get p = 3.

Verification / Alternative check:
If p = 3, both equations become 3 x + y = 2 and 3 x + y = 4. These clearly have the same left side 3 x + y and different constants, so they represent two distinct parallel lines with equal slope and different intercepts. This confirms that p = 3 is consistent with the geometric meaning of parallel lines.

Why Other Options Are Wrong:
- Option 1: If p = 1, the first line slope becomes -1, which does not match slope -3 of the second line.- Option 2: If p = 2, the first line slope is -2, still not equal to -3.- Option 4: If p = 4, the first line slope is -4, again different from -3.- Option 5: p = 5 gives slope -5, which also fails the parallelism condition.

Common Pitfalls:
Many learners incorrectly compare constant terms or think that for parallel lines all coefficients must be proportional, including the constant term. For parallel but distinct lines, only the ratios of the coefficients of x and y matter, not the constant term. Another common mistake is to forget that the slope for a x + b y = c is -a / b and to reverse a and b. Moving terms to slope intercept form y = m x + c helps clarify this. Always check that you are equating slopes, not intercepts, when working with parallel lines.

Final Answer:
The value of p that makes the two lines parallel is 3.