Let's go through the questions one by one:

1. Finding the Demand Function:

The problem states that the function passes through points (3,3) and (2,4). To find the linear function $P = mx + c$:

• The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 3}{2 - 3} = -1$
• Using the point-slope form: $P - y_1 = m(x - x_1)$: $P - 3 = -1(x - 3) \implies P = -x + 6$ So, the correct answer is B. $P = -x + 6$.

2. Finding the Intersection Point:

Given equations $Y = 2 + 2x$ and $Y = 10 - 2x$:

• Set the equations equal to each other: $2 + 2x = 10 - 2x$
• Combine like terms: $4x = 8 \implies x = 2$
• Substitute $x = 2$ into one of the original equations: $Y = 2 + 2(2) = 6$ Thus, the intersection point is (2, 6), so the correct answer is B. (2, 6).

3. Finding the Slope of Line AB:

Given points $A(2, 8)$ and $B(4, 5)$:

• The slope $m$ is: $m = \frac{5 - 8}{4 - 2} = \frac{-3}{2} = -1.5$ So, the correct answer is C. -1.5.

4. Equation of a Line Through (4,1) with Slope 2:

• Using point-slope form: $Y - y_1 = m(x - x_1)$: $Y - 1 = 2(x - 4) \implies Y - 1 = 2x - 8 \implies Y = 2x - 7$ So, the correct answer is C. $Y = 2x - 7$.

5. Finding the Value of $a$ for Parallel Lines:

• The line $Y = ax + 2$ needs to be parallel to the line passing through $(2,4)$ and $(3,1)$.
• The slope of the line through these points is: $m = \frac{1 - 4}{3 - 2} = \frac{-3}{1} = -3$
• Since parallel lines have the same slope, $a = -3$. So, the correct answer is A. -3.

Would you like more detailed explanations or have further questions?

Additional Questions for Further Practice:

1. How do you find the slope of a line given two points?
2. What is the general equation of a line in slope-intercept form?
3. How can you determine if two lines are parallel or perpendicular?
4. What is the process for finding the intersection of two linear equations?
5. How can you verify if a point lies on a given line?

Tip:

When determining the slope between two points, always subtract the coordinates in the same order (either always second point minus first or vice versa) to maintain consistency.