The document contains several multiple-choice math questions related to linear equations, slopes, and points on a graph. Let's go through the questions one by one and provide solutions:

1. Find the demand function that passes through the points (3,3) and (2,4):

   To find the equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$:

   • Slope ($m$) = $\frac{y_2 - y_1}{x_2 - x_1}$ = $\frac{4 - 3}{2 - 3} = -1$.
   • Equation of the line in slope-intercept form: $y = mx + c$.
   • Using one of the points (3,3) to find $c$: $3 = -1 \cdot 3 + c$ ⟹ $c = 6$.
   • Therefore, the equation is $y = -x + 6$.

   Correct Answer: B. $P = -x + 6$

2. Find the intersection point of the lines $Y = 2 + 2x$ and $Y = 10 - 2x$:

   Set the equations equal to each other to find the intersection point: $2 + 2x = 10 - 2x$, $4x = 8$, $x = 2$, Substitute $x = 2$ into one of the equations to find $y$: $Y = 2 + 2 \cdot 2 = 6$.

   Intersection Point: $(2, 6)$.

   Correct Answer: B. (2,6)

3. Given points $A(2,8)$ and $B(4,5)$, find the slope of line $AB$:

   Slope ($m$) = $\frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 8}{4 - 2} = \frac{-3}{2} = -1.5$.

   Correct Answer: D. -1.5

4. Equation of the line passing through point $(4,1)$ with slope $2$:

   Use point-slope form: $y - y_1 = m(x - x_1)$, $y - 1 = 2(x - 4)$, $y = 2x - 8 + 1$, $y = 2x - 7$.

   Correct Answer: C. $Y = 2x - 7$

5. Find the value of $a$ so that the line $Y = ax + 2$ is parallel to the line passing through points $(2,4)$ and $(3,1)$:

   Slope of the line passing through $(2,4)$ and $(3,1)$: $m = \frac{1 - 4}{3 - 2} = -3$, Parallel lines have equal slopes: $a = -3$.

   Correct Answer: A. -3

Would you like further details or have any questions about these solutions?

Related Questions:

1. What are the properties of parallel lines in a coordinate plane?
2. How can the slope-intercept form of a line be derived from its standard form?
3. What is the geometric interpretation of the slope of a line?
4. How do you determine if two lines are perpendicular using their slopes?
5. How can the equation of a line be determined given a point and a slope?

Tip:

When determining if two lines are parallel, remember that their slopes must be equal. For perpendicular lines, their slopes are negative reciprocals of each other.