To determine the value of [tex]\( m \)[/tex] for which the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is empty, let's analyze the given information:
1. [tex]\( U \)[/tex] is the set of all points on the coordinate plane.
2. [tex]\( A \)[/tex] is the set of points that are solutions to the equation [tex]\( y = 2x + 5 \)[/tex].
3. [tex]\( B \)[/tex] is the set of points on the line [tex]\( y = mx \)[/tex].
We need to find the value of [tex]\( m \)[/tex] such that the set intersection [tex]\( A \cap B = \varnothing \)[/tex], which means the lines represented by [tex]\( A \)[/tex] and [tex]\( B \)[/tex] do not intersect.
### Step-by-Step Solution:
1. Understand the equations of the lines:
- The line representing set [tex]\( A \)[/tex] is described by the equation [tex]\( y = 2x + 5 \)[/tex].
- The line representing set [tex]\( B \)[/tex] is described by the equation [tex]\( y = mx \)[/tex].
2. Intersection condition:
- Two lines on a coordinate plane will not intersect if they are parallel and distinct. Parallel lines have the same slope but different y-intercepts.
3. Determine when the lines are parallel:
- For the line [tex]\( y = 2x + 5 \)[/tex] (slope [tex]\( 2 \)[/tex]) and the line [tex]\( y = mx \)[/tex] to be parallel, their slopes must be equal.
- Therefore, the slope of [tex]\( y = mx \)[/tex] must be 2 for them to be parallel. In this case, the equation for line [tex]\( B \)[/tex] becomes [tex]\( y = 2x \)[/tex].
Since we are looking for the value of [tex]\( m \)[/tex] that makes the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] empty, and we determined that the two lines will not intersect when they are parallel but distinct, [tex]\( m \)[/tex] needs to be 2.
Thus, the value of [tex]\( m \)[/tex] for which [tex]\( A \cap B = \varnothing \)[/tex] is:
[tex]\[ m = 2.0 \][/tex]
So, [tex]\( m = 2.0 \)[/tex].
